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Setup of a Laser System for Ultracold Sodium - Towards a

1. 2 1 Dye Lasers aE Three stage Birefringent Filter Birefringent Plate Optical Axis Rotation Axis Laser Beam Figure 2 5 Schematic of a three stage birefringent filter On the left one of these stages is drawn range is given by c Af d 2 8 n Nolld It is evident that for all other wavelengths the outgoing polarization is elliptical or in special cases circular or linear yielding losses between 0 and 100 at the polarizer The overall transmission is given by 58 T f d cos HE 2 9 where fo denotes some frequency with T fo 1 within the relevant frequency range Concatenating several Lyot filters to the aforementioned three staged bire fringent filter results in narrower transmission peaks while conserving the free spectral range See figure 2 6 for plots of single and combined transmission curves The assembly discussed so far does not allow for wavelength tuning Further more one can realize such a filter more easily in the case of a dye laser or any laser with a homogeneously broadened gain medium As discussed in section 2 1 1 a very small percentage of additional losses is sufficient for the suppression of a mode in such a laser Polarizers can be replaced by a plate under Brewster s angle Refer to figure 2 5 for an overview The plates are no longer perpendicular to the incident beam but are mounted rotatably and under Brewster s angle Brewster surfaces serve as polarizers introducing
2. 47 Chapter 3 Experimental Setup atoms can be increased by a factor of ten by blocking the repumping light in the center of the MOT By chance we found out that we can create a close to perfect ring shaped mode by sloppily coupling into the multi mode fiber Focusing the laser beam into the incoupler instead of collimating the beam obviously excites whispering gallery type mode within the fiber cladding The width of the ring structure can easily be tuned by displacing the focus relative to the incoupler The incoupling exceeded 50 without any further optimization If ever the core of the mode is not dark enough an additional aperture can be inserted without losing much power The outcoupled structure is shown in figure 3 7 The ring is covered with a speckle pattern even though we do not think that this will interfere one could remove it by shaking the fiber The slightest movement of the fiber rotates the pattern thus a small loud speaker should suffice 20Spatial filtering is an alternative but breaks a butterfly on a wheel 48 3 5 The Laser System MOT beam Imaging MOT To Zeeman to fiber splitter Repumper slower O O O ya 1 2 2 0 2 2 2 2 2 3 5 ied DYE LASER N IR 32 Nog a p IN IN N Ln LN lt 3 1 4 RF coils Sodium Spectroscopy NX 9 A A Polarizing Lens for Beam Ph i r oto Fiber Window Cat s Eye Dump Diode Coupler Mirror
3. Chapter B Atomic Beam Shutter B 5 Source Code list p 12f683 5 ea a ak ae R SR SR SK SR SR SR ak K ak K ae SR SR SR I I R R ae SR SR SR SR SR SK I A R SR SK SR SR SR SR SR ke kk K fe ae SR SR SR SR SR SR 3K k k K ak ak ak Driver for atomic beam shutter of the NaLi Experiment O Kirchhoff Institut f r Physik University of Heidelberg AG Oberthaler Kk K KK k k Connect a standard servo to Pin 5 PWM driven duty cycle 5 10 50Hz period Connect voltage dividers to Pins 3 and 6 Connect a Status LED to Pin 7 I_max 25mA Connect OV to Pin 1 Connect 5V to Pin 8 We ee ee s Choose the tvo setpoints by means of the voltage dividers The servo position may nov be selected vith TTL Signals on Pin 2 ak ak R R R ak ka SK SK K ak SR SR SR SR 3K 3K R 3K R K I I I I SR SR II SR SR SR R R SR R K SK SK a SSR SR SR SR SR R OR SR R R K R K SR aa K K K K pin assignment AAA he Pin 7 GPO gt out 0 LED Status out Pin 6 GP1 gt in 1 A D in Pin 5 GP2 gt out 2 PWM out Pin 4 GP3 gt in 3 MCLE in Pin 3 GP4 gt in 4 A D in Pin 2 GP5 gt in 5 TTL in Pin 1 OV Pin 8 5V V_ref for A D Conversion ak ak ak ak k K SK K ak SR SR SR SR R R SR R K I I SSR I I I II SR SR SR 3K aI R I I a I SK SR 4 K SR SR SR aa R a a ae ak ak ak a K SR SR Ok K K K ak Version September 30th 2007 aut
4. D S Durfee D M Kurn and W Ketterle Bose Einstein Condensation in a Gas of Sodium Atoms Phys Rev Lett 75 22 3969 Nov 1995 W Pauli The Connection Between Spin and Statistics Phys Rev 58 716 722 1940 L V de Broglie Recherches sur la Th orie des Quanta PhD thesis Sorbonne Paris 1924 R Gati B Hemmerling J F lling M Albiez and M K Oberthaler Noise Thermometry with Two Weakly Coupled Bose Einstein Condensates Physical Review Letters 96 13 130404 April 2006 M Albiez R Gati J F lling S Hunsmann M Cristiani and M K Oberthaler Direct Observation of Tunneling and Nonlinear Self Trapping in a Single Bosonic Josephson Junction Physical Review Letters 95 1 010402 June 2005 79 Bibliography 10 K Xu Y Liu J R Abo Shaeer T Mukaiyama J K Chin D E Miller W Ketterle K M Jones and E Tiesinga Sodium Bose Einstein condensates in an Optical Lattice Physical Review A 72 4 043604 October 2005 11 D Rychtarik B Engeser H C Nagerl and R Grimm Two Dimensional Bose Einstein Condensate in an Optical Surface Trap Physical Review Let ters 92 17 173003 2004 12 M R Matthews B P Anderson P C Haljan D S Hall C E Wieman and E A Cornell Vortices in a Bose Einstein Condensate Phys Rev Lett 83 13 2498 Sep 1999 13 B Demarco and D S Jin Exploring a quantum degenerate gas of fermionic atoms Physical Review 58 4267 December 1998 14 B De
5. Figure 3 8 Overview of our sodium laser setup the transfer beam for the magnetic trap has not yet been implemented MOT Zeeman MOT Imaging to fiber splitter Slower Repumper O O 1 2 1 2 2 1 2 242 242 242 242 s n 1 de DYELL ER yy EN T LN CNR 5 1 Zeeman 77 1 1 1 1 Repumper ID Alm 44 4 214 M4 2 4 M2 E Lock In Amplifier Y II I A En 1 pip ci M Figure 3 9 Draft of our lithium laser setup 49 Chapter 3 Experimental Setup 50 4 First Measurements This chapter will give the first absorption measurements we have performed on our magneto optical trap for sodium More extensive studies are part of Marc Repp s diploma thesis 72 Here we present the first absorption measurements on our MOT All results we obtained have to be considered preliminary since this work has still been done using some provisional coils In addition systematic optimization of the different laser detunings intensities and magnetic fields have not yet been done One change compared to Marc Repp s diploma thesis is that the 1 on 6 fiber splitter has been installed in the meantime However the number of trapped atoms was still up to a factor of ten lower than before 4 1 A Provisional Absorption Imaging System 4 1 1 Optical Density of an Atomic Cloud and the Beer Lambert Law Light cros
6. Shaping the magnetic field B z such that deceleration i e Doppler shift of the atomic lines is compensated by Zeeman shifts of the used 3A slower designed for lithium only would have been shorter while maintaining the same slowed fraction of slowed atoms 36 3 3 Zeeman Slower transitions the atoms are resonant all along the magnetic field and thus slowed down Atoms moving at velocity v in a position dependent magnetic field B z with a counterpropagating laser beam with wavenumber k experience a deceleration T uc 7 5 2 5 3 1 m has 229 gt B 5 z v y Ba 12970 3 2 where h denotes the reduced Planck constant T is the line width of the transi tion of the excited atoms the Land factor g for the used transition is very close to one 64 65 m is the atomic mass and S is the saturation parameter refer to 59 for more details on atom light interaction o is called laser detuning and is given by the difference between the laser and the atomic resonance frequency Elimina tion of v while keeping a constant for all z implies a square root shaped magnetic field Main characteristics of a Zeeman slower are the S parameter needed for slowing in our case S should be on the order of 1 the maximal velocity that can be slowed in our case about 700 m s for both species and the configuration of the magnetic fields We set up a slower in a so called zero crossing configuration also known as spin flip slower 7
7. This influences transition amplitudes within the level scheme in such a way that an additional cooling mech anism arises leading to more efficient cooling referred to as sub Doppler cooling 166 671 Temperatures on the order of 104K can be achieved for sodium corre sponding to the kinetic energy associated with a single photon momentum recoil This is referred to as the recoil limit Already at that point it becomes clear that sub Doppler cooling will not work for lithium see figure 3 4 The excited states are separated by less than the 30 2 2 Magneto Optical Trapping natural linewidth thus selective excitation of only one F state is not feasible 2 2 5 Repumping One point has not been mentioned yet According to equation 2 14 electrons in sodium may also be excited to the F 2 state since the level spacing is not infinite Thus transitions to the F 1 state are no longer dipole forbidden Once an atom is in the F 1 state it is no longer trapped and needs to be transferred back to the F 2 state Therefore a repumping beam resonant with the F 1 to F 2 or F 1 transition is added This effect is even more pronounced in lithium due to the quasi degeneracy of the excited states See 3 5 2 for more details 2 2 6 Limitations and the Dark Spot MOT for Sodium The density in a sodium MOT is limited to about 10 em mainly by a process called radiation trapping 68 Already at much lower densities the atoms are no longer int
8. k ak ak k ak k k k ak k ak k ak k ak k ak ak k k ak ak k ak ak k k ak k ak ak ak k ak ak k ak ak k ak ak k ak k ak ak ak ak ak kok Init CLRF BANKSEL bsf bef bef bsf Flags OSCCON OSCCON IRCF2 OSCCON IRCFi OSCCON IRCFO OSCCON SCS set internal frequency to 1MHz gt sactivate internal clock SET PIN USAGE I 0 analog digital BANKSEL CLRF BANKSEL MOVLW MOVWF BANKSEL CLRF MOVLW MOVWF BANKSEL MOVLW MOVWF GPIO GPIO TRISIO Ox3A TRISIO ANSEL ANSEL Ox7A ANSEL ADCONO 0x05 ADCONO CONFIGURE PWM module BANKSEL bsf bsf BANKSEL MOVLW MOVWF BANKSEL bsf bsf bef return T2CON T2CON T2CKPS1 T2CON MR2ON PR2 OxFF PR2 CCP1CON CCP1CON CCP1M3 CCP1CON CCP1M2 CCP1CON CCP1M1 send of initialization Init GPIO to zero Set GP lt 0 2 gt as outputs and set GP lt 1 3 5 gt as inputs digital 1 0 Set GP lt 1 4 gt i e AN lt 1 3 gt as analogue inputs use internal clock for conversion left justified 8 MSBs in ADRESH A D Conversion on prescaler Timer2 started PR2 Timer2 prescaler gt 50Hz 1MHz Clock PWM chosen in mode active high gt k k k k k ak k k k k k k ak ak ak ak k ak ak k ak ak ak k ak ak k k ak k ak k ak k ak k ak ak ak k ak ak k ak ok k k ak k ak k k k k k k k k k ak ak k ak ak k k ak k ak ak ak 5
9. 140 Radius of Solenoid 12 5 mm Length of Solenoid 55mm Inductance measured theoretical 0 17mH 0 18mH Resonance Frequency with 27 nF in series 74 2 kHz Peak magnetic Field at resonance 6G Table D 1 Characteristics of the RF coil using a Voltcraft 7202 wobbel function generator 73 Chapter D Spectroscopy Cell and Doppler Free Laser Locking 74 scanning laser frequency f t df dt lt 0 df dt gt 0 0 21 a b c 3P a D 3S F 2 3S F 1 Time t A U Figure D 5 D line of sodium with the transitions a 35 2 F 2 gt 3P3 2 b 351 2 F 1 3P y and the cross over line The dashed line completes the Doppler profile of the line schematically mn nn Co N Error Signal A U 0 100 200 300 400 500 600 700 frequency A U Figure D 6 Error signal i e derivative of the transition 35 2 F 2 gt 3P3 with partially resolved hyper fine structure of the excited state Locking is done on the cross over peak of the F 2 F 2 F 3 transitions b a F 2 F 3 c cross over F 2 F 1 F 3 d F 2 FP 2 e cross over F 2 F 1 F 2 e F 2 gt PF 1 E RF Drivers for High Frequency Components For driving the high frequency AOM at 1859GHz and the EOM at 1713GHz a drivers have been set up A schematic can be found in figure E 1 containing a list of levels attenuations and
10. 314 Jul 1986 62 J D Jackson Klassische Elektrodynamik Gruyter 2002 63 S Chu L Hollberg J E Bjorkholm A Cable and A Ashkin Three Eimensional Viscous Confinement and Cooling of Atoms by Resonance Radi ation Pressure Phys Rev Lett 55 1 48 Jul 1985 64 D A Steck Sodium D Line Data available at http steck us alkalidata 2003 65 M A Gehm Properties of 6Li extract from PhD thesis 2003 66 P D Lett R N Watts C I Westbrook W D Phillips P L Gould and H J Metcalf Observation of Atoms Laser Cooled below the Doppler Limit Phys Rev Lett 61 2 169 Jul 1988 67 J Dalibard and C Cohen Tannoudji Laser Cooling Below the Doppler Limit by Polarization Gradients Simple Theoretical Models J Opt Soc Am B 6 11 2023 1989 68 T Walker D Sesko and C Wieman Collective Behavior of Optically Trapped Neutral Atoms Phys Rev Lett 64 4 408 Jan 1990 69 W Ketterle K B Davis M A Joffe A Martin and D E Pritchard High Densities of Cold Atoms in a Dark Spontaneous Force Optical Trap Phys Rev Lett 70 15 2253 Apr 1993 70 Z Hadzibabic S Gupta C A Stan C H Schunck M W Zwierlein K Dieck mann and W Ketterle Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms Physical Review Letters 91 16 160401 Octo ber 2003 71 Z Hadzibabic Studies of a Quantum Degenerate Fermionic Lithium Gas PhD thesis Massachusetts Institute of T
11. Mixtures Phys ical Review Letters 99 13 130407 2007 81 Bibliography 35 36 37 38 39 40 41 42 43 44 45 82 T Bourdel L Khaykovich J Cubizolles J Zhang F Chevy M Teichmann L Tarruell 5 Kokkelmans and C Salomon Experimental Study of the BEC BCS Crossover Region in Lithium 6 Physical Review Letters 93 5 050401 2004 C A Stan M W Zwierlein C H Schunck S M Raupach and W Ketterle Observation of Feshbach Resonances between Two Different Atomic Species Physical Review Letters 93 14 143001 sep 2004 C Ospelkaus S Ospelkaus L Humbert E Ernst K Sengstock and K Bongs Ultracold heteronuclear molecules in a 3d optical lattice Physical Review Letters 97 12 120402 2006 K B Davis M O Mewes and W Ketterle An Analytical Model for Evapo rative Cooling of Atoms Appl Phys B 60 155 1995 M D Barrett J A Sauer and M S Chapman All Optical Formation of an Atomic Bose Einstein Condensate Phys Rev Lett 87 1 010404 Jun 2001 C J Myatt E A Burt R W Ghrist E A Cornell and C E Wieman Produc tion of Two Overlapping Bose Einstein Condensates by Sympathetic Cooling Phys Rev Lett 78 4 586 Jan 1997 W Zhang C A Sackett and R G Hulet Optical Detection of a Bardeen Cooper Schrieffer Phase Transition in a Trapped Gas of Fermionic Atoms Physical Review A 60 504 July 1999 Z Hadzibabic C A S
12. Trapping 25 2 2 1 Light Forces on Two Level Atoms 25 Contents 22 11 Bipole Poree YEAR AAA A Oa 26 2 2 1 2 Light Pressure Force 4 4a 2 4 x ar 22 27 2 2 2 Optical Molasses ar a ae 27 2 2 3 Magneto Optical Trapping of Multilevel Atoms 28 2 2 4 Sub Doppler Cooling 4 0 2 4 2 2 8322 aa R a 30 2 2 90 Repumpine ados pane r greg a a da Sb gs ee 31 2 2 6 Limitations and the Dark Spot MOT for Sodium 31 3 Experimental Setup 33 3 1 IAE ae pz l y ie Eag 33 Vacuum Oystem A ee Ae o bo EEE 34 3 2 1 The Vacuum Chamber sas oS gel XA 34 3 222 IO ek ee eae ok cee ge oe Sa we ae Ware Pe 35 33 Leeman Slower ar ee a te Sie Be 35 Desigm COMA s 24 22 as olen ah A a 36 3 3 2 Setup and a Basic Introduction 36 3 4 Magnetic Fields 2 4 44 sa Bae hd Be ad Meee he 38 3 4 1 Feshbach Coils 7 2 ua 200 oR a me a 38 Magnetig Trap al wa Br ae pu za 39 The Laser Systems eiras rm Hensel 40 3 5 1 Why Dye Lasers s Bee ne Sik ee 40 335 Drequeneies Mies Es a ach 40 3 5 2 1 Locking the Laser to an Atomic Resonance 41 339722 MOL aaa Bre A ada b s S M S eee aie 42 3 5 2 3 MOT Repumper 43 3 9 2 4 Zeeman Slower 43 3 5 2 5 Zeeman Slower Repumper 44 359 2 0 HMAC Sue e AA AAA 44 3 5 2 7 Transfer into a Magnetic Trap 4
13. about 7W of pumping power for 6 bar of dye pressure However stable single mode operation is only possible up to about 6 W 18 2 1 Dye Lasers M3 thin quartz plate output M4 OC dye jet pump beam Mp M1 Figure 2 4 Overview on our ring dye laser source Manual for Radiant Dyes Ring Dye Laser As a pump laser we use a Radiant Dyes MonoDisk laser which is a frequency doubled diode pumped Yb YAG ring laser emitting two laser beams of gt 10W each at 515nm This is due to the absence of an optical diode enabling two counter propagating beams to persist The active medium is inhomogeneously broadened permitting two different non competing longitudinal modes One of the two outputs is focused onto the dye jet by Mirror Mp Mirrors M1 M2 M3 and M4 are forming the cavity M1 and M2 are concave mirrors with a radius of curvature r 150mm focusing the intracavity laser beam onto the dye jet 2 1 3 1 Optical Diode OD and Thin Quartz Plate In order to avoid competing counterpropagating beams an optical diode also called Faraday isolator is inserted into the cavity introducing additional losses for beams directed from M3 to M4 All optical elements made of glass except for the etalons are brought into the cavity under Brewster s angle This induces losses for modes with polarization perpendicular to the laser plane The quartz plate i e A 2 plate rotates the polarization by 45 Within the Fara
14. cooling process using the bosons as a refrigerant for the fermionic component An important advantage is that the atom number of fermions decreases only slightly 1 3 1 General Aspects During the last few years several groups have already reached Fermi degeneracy with different approaches and various combinations of elements 14 41 42 43 This section is meant to be an overview on different isotopes used together with a brief discussion of their pros and cons Common to nearly all of these experiments is that either Li or K are used Alkalis have in common that their level schemes are simple and well understood Li and K half life time 10 years are the only stable alkaline fermionic iso topes Both show Feshbach resonances see section 1 2 1 for details on Li at reasonable magnetic fields However Li offers a resonance with a width of about 100G whereas in K the widths on the order of one Gauss can hardly be resolved 45 Another point in favor of Li is that molecules formed in the vicinity of a Feshbach resonance have higher lifetimes 46 47 On the other hand K has a resolved hyper fine structure in the excited state thus better laser cooling is pos sible see section 2 2 4 Advantageous about K is also that its vapor pressure is much higher at a given temperature than for Lithium The magneto optical trap cf 2 2 can thus be loaded from the background pressure created by a small dispenser whereas a
15. filters have been removed The software for MATLAB captures images from the camera continuously fits and plots them and provides further function ality described in the following section C 1 Application Notes C 1 1 Warnings This program uses the MATLAB Image Acquisition Toolbox make sure this add on is installed furthermore the application is relatively performance consuming In order to obtain a reasonable frame rate use an up to date PC This is not a high precision solution even though some comparisons to more precise methods have demonstrated accuracies of better than 5 if the fol lowing precautions are met Whenever the chip is close to saturated waists are overestimated So try to keep gray values below 150 The CCD and the absorber in front do not sustain infinitely much intensity make sure to reduce the power contained in the beam to a reasonable value Whenever you execute the live_ BeamProfiler m file the camera driver enables the property Mich verfolgen within the control panel This feature tries to center somebody s face onto the captured images so pretty useless in this Built by Tobias Schuster 67 Chapter C Beam Profiler case However it seems to be impossible to change this default setting so do not forget to uncheck it C 2 Functionality C 2 1 Overview Graphs and Fitting After connecting the beam profiler and if required installing the driver software start the live Beam
16. for any stray light and camera noise The logarithm of the point by point ratio of the resulting two images gives the optical density per pixel 53 Chapter 4 First Measurements 4 2 Estimating the Atom Number in the Sodium MOT The diameter of the MOT we have analyzed was on the order of 2mm The magnetic field gradient of the quadrupole field was below 10G cm thus Zeeman shifts of the atoms are on the order of 2 MHz and thus negligible Doppler effect well below 1MHz does not play a role either Images have been taken with the magnetic quadrupole field still on For an estimate of the atom number any absorption properties depending on the quantization axis have not been taken into account in the following and the minimal saturation intensity has been used for calculations for sodium Isat 6 2 mVV em for the transition F mp F 12 2 13 3 Hence the absorption image can directly be identified as the density distribution projected onto the CCD At the same time the number of particles is underestimated systematically 0 4 w 0 2 Optical Density Optical Density 4 0 1 2 3 4 5 6 Vertical Position um Vertical Position um a b Figure 4 2 Averaged column density profile a and row density profile b shown in figure 4 1 together with Gaussian fits Figure 4 1 shows one of the first images taken in figure 4 2 the averaged row and column density
17. high temperature oven needs to be used for Li Once the fermionic part is chosen the choice of bosons is reduced taking into account that for optimal heat transfer the masses of the two species should not 10Recently the group of Yoshiro Takahashi achieved Fermi degeneracy with exotic Ytterbium Yb offering two stable fermionic and five stable bosonic isotopes all with reasonable natural abundance 44 We will restrict our discussion to the aforementioned elements They are hydrogen like with only one electron in the outermost shell data from http atom kaeri re kr ton 10 1 4 Outline differ too much The same condition holds for magnetic trapping cf section 3 4 2 as for different masses the centers of the trap for the different species are shifted slightly due to the gravitational force also called gravitational sag This impairs their heat contact The bosonic counterpart should not be lighter than the fermions As seen in equation 1 6 the critical temperature decreases for increasing mass As a consequence the deeper one wants to cool into degeneracy the lighter the fermion should be relative to the boson Additionally one needs a sufficiently large thermal bath for the desired size and temperature of the fermionic sample Finally the inter species collisional properties are of importance however they are generally not predictable and have to be acquired experimentally Resuming the previous arguments mainly fo
18. is only about 3cm s for sodium and lithium the big linewidths of T 27 10 MHz lead to accelerations on the order of lal 10 m s Yscatter 2 14 2 2 2 Optical Molasses Up to now we have neglected any effects arising from the movement of the atoms Moving atoms experience a Doppler shifted light frequency and thus a velocity This force is also known as scattering force radiation pressure force and dissipative force 10Containing the laser detuning but also Zeeman or Doppler shifts 27 Chapter 2 Theory dependent detuning In this case equation 2 14 needs to be modified slightly by redefining 6 to be A k v where A denotes the laser detuning and the second term corresponds to the Doppler shift Given two red detuned laser beams in opposite directions atoms moving in either direction are shifted into resonance with the counterpropagating beam and slowed down Adding two more pairs of beams in the other spatial dimensions achieves efficient cooling This setup is called optical molasses the atoms behave like in a highly viscous fluid Cooling to zero temperature is of course not achieved On average atoms are emitting photons of lower frequency than they are absorbing The difference heats up the sample and cancels the cooling effect at some point The corresponding temperature is referred to as Doppler temperature This cooling scheme is called Doppler cooling and yields temperatures on the order of hun
19. k k k gt k ak ak ak k k ak k ak ak ak ak ak k ak ak ak k ak ak k ak k k k ak k ak k ak k ak k ak ak k k ak ak k ak ak k k ak k ak ak ak k k k k ak ak k ak ak k ak ak k k ak ak ak kok Main Main call Init goto chgd_to_high loo BANKSEL GPIO btfsc GPIO 5 goto chgd_to_high goto chgd_to_low chgd_to_high BANKSEL STATUS btfsc Flags 0 goto loop BANKSEL ADCONO in order to enable PWM on startup get TTL Signal 12 high go to chgd to high else go to chgd to lov sif Flag bit is also equal to one return to loop no change 65 Chapter B Atomic Beam Shutter bsf ADCONO CHS1 bsf Flags 0 bsf GPIO 0 goto Changed chgd_to_low BANKSEL STATUS btfss Flags 0 goto loop bef ADCONO CHS1 bef Flags 0 bef GPIO 0 goto Changed Changed NOP BANKSEL STATUS BSF ADCONO GO BTFSC ADCONO GO GOTO 1 BANKSEL ADRESH we ve s we RRF ADRESH 1 RRF ADRESH 1 bef ADRESH 6 bef ADRESH 7 movlw 20 ADDWF ADRESH 1 SWAPF ADRESH movwf temp bsf temp 3 bsf temp 2 bcf temp 1 MOVFW tem MOVWF 1 rrf ADRESH 1 rrf ADRESH 1 bef ADRESH 7 bef ADRESH 6 movfw ADRESH movuf CCPR L goto loop 66 choose A D Channel wait for about 5 microseconds i e 5cycles until multiplexing has been finished set Flag bit to current TTL level 1 set Status indicator LED to high on else run Changed function 12 F
20. lengths in the vicinity of Feshbach resonances only one single pair of coils are used creating a quadrupole field in an anti Helmholtz configuration or a homogeneous field at the center for a Helmholtz configuration depending on the relative polarity of the coils Here the Feshbach coils are the challenging part They will be described in the following Figure 3 2 Drawing of our multi functional coils used for magneto optical and magnetic trapping as well as for Feshbach fields 3 4 1 Feshbach Coils Tuning over atomic Feshbach resonances of a degenerate Fermi gas opens up a wide field of exciting physics that becomes accessible as described in the introduction see chapter 1 2 In a standard experiment interaction is tuned to some scattering length in the range of the Feshbach resonance i e a magnetic field of about 830 G for Li Typically the interaction needs to be switched very quickly Our coils are designed such that they can be switched from 1000 G to zero in less than 20 us For a minimal switching time and bearable inductive voltages the inductance needs to be kept as low as possible So in order to reach the high magnetic fields huge currents up to 440A in our case have to be put through the wire leading to dissipated powers on the order of 2kW Heat management is an important point since any temperature instability causes fluctuations of the magnetic field gradient 38 3 4 Magnetic Fields In this kind of experiment on
21. n to observe effects arising mainly from quantum statistics Therefore we are inserting equation 1 4 into equation 1 5 Using that the particle density n in an ideal gas is given by n P kgT where P denotes the pressure yields a critical temperature on the order of h2n2 3 2mkp C 1 6 This temperature is referred to as critical temperature for bosons and Fermi tem perature for fermions Yet for a first estimate equation 1 6 is sufficient Typical densities in experiments with cold atoms are on the order of 1014 atoms cm The molar mass of Li is 6g mol This yields temperatures on the order of 1 uK Comparing this to a degenerate gas of electrons in a metal gives rise to a factor of 104 mass ratio plus another six orders of magnitude density ratio 1092 9 thus an overall factor of about 1016 in temperature equivalent to temperatures on the order of 10000K So one could ask Why should one be interested in such hard to create systems We would like to motivate this in the next section 2A more accurate analysis yields another factor of 2 612 in phase space density for spinless atoms resulting in temperatures that are about a factor of two lower Corrections due to interactions turn out to be small Chapter 1 Introduction 1 2 Degenerate Fermions Bose Einstein condensation has been a very active field of research during the last 12 years In the meantime there are approximately 60 BEC experiments in the wo
22. nen und lehrreichen Stunden im Labor und aufer halb der gesamten Arbeitsgruppe f r die Hilfe die ich w hrend des Jahres erfahren habe die unterhaltsamen Kaffeepausen und die gemeinsamen Abende beim Grillen Beachvolleyball und Karten spielen dem NaLi Team besonders aber Peter Kr ger und Jan Krieger f r das Kor rekturlesen meiner Arbeit Mein Dank geht auch an die elektronische und mechanische Werkstatt des Instituts insbesondere an Herrn Spiegel und Herrn Herdt sowie die Glas bl serei des Physikalischen Instituts meinen Eltern die mich immer unterst tzt haben und f r mich da waren Ohne Euch w re dieses Studium so nicht m glich gewesen 1 www wikipedia de www wikipedia org 77 KapitelF Danksagung e meiner Freundin Iris f r die Geduld Unterst tzung und die wundervolle Zeit 78 Bibliography u 2 3 4 5 6 7 8 9 S N Bose Plancks Gesetz und Lichtquantenhypothese Z Phys 26 178 1924 A Einstein Quantentheorie des Einatomigen Idealen Gases Kgl Preuss Akad Wiss 22 178 1924 A Einstein Quantentheorie des Einatomigen Idealen Gases Zweite Abhand lung Sitzungsber Kgl Preuss Akad Wiss page 3 January 1925 M H Anderson J R Ensher M R Matthews C E Wieman and E A Cor nell Observation of Bose Einstein Condensation in a Dilute Atomic Vapor Science 269 5221 198 July 1995 K B Davis M O Mewes M R Andrews N J van Druten
23. off the center of the magnetic quadrupole field the cycling transition is shifted into resonance with a counterpropagating beam leading to a backward force This is visualized in figure 2 11 The motion of atoms in a MOT is now characterized by a spring constant attributed to the Zeeman shifting and a damping coefficient arising from the optical molasses Since damping is much bigger than the spring constant the atomic motion is overdamped 29 Chapter 2 Theory o Light Magnetic field Position Figure 2 11 Trapping schematic in a MOT o and o denote the polarization of the light beams coming from the left and right respectively wo denotes the unshifted resonance frequency w is the frequency of the laser beams that is slightly detuned to the red Read the x axis to be the magnetic field or in the case of a MOT with B x x z also as the position in a trap For weak magnetic fields the Zeeman substates are shifted such that Azeeman gr mp grmp uBB 2 with F 3 F 2 mp 3 mp 2 gp 0 6671 and gr 0 5006 grmp 1 0002 1 64 An atom on the right will mainly absorb photons from the o beam and vice versa pushing the atoms to the center 2 2 4 Sub Doppler Cooling The standing light wave formed by two counter propagating circularly polarized beams is linearly polarized at each point The polarization vector rotates along the beam with a periodicity of half the wavelength
24. selecting the temperatures in the two oven chambers About 10cm behind the oven nozzle there is a rotary feedthrough 5 inserted into our vacuum system This feedthrough is connected to a thin plate made of stainless steel serving as a shutter for the atomic beam During this diploma thesis a driver has been developed that allows to switch the atomic beam on and off with TTL signals Basically this driver uses a standard RC servo and is controlled by a small PIC microcontroller See appendix B for more details A small aperture at position 6 is blocking atoms on off axis trajectories The transmitted atomic beam is sent through two differential pumping tubes 7 and 8 permitting to decrease the residual gas pressure gradually from 5 1078 mbar to 107H mbar We will present the Zeeman slower 9 in the next section The Zeeman slower is put as close as possible to the glass cell avoiding too many losses due to the elevated divergence of the beam at that point The glass cell 10 itself is attached to flexible bellow adapters via a glass to metal transition Behind the glass cell there is a vacuum gauge 11 and a window for the Zeeman slower beam 12 3 2 2 Pumping Our vacuum setup is evacuated by two 551 s ion getter pumps one 13 in the oven and collimation section the other one 14 in between the two differential pumping tubes and a 1501 s ion getter pump 15 combined with a titanium sublimation pump about 10001 s behind the
25. this experiment a vacuum apparatus with a residual pressure well below 10710 mbar is needed This is for several reasons The mean free path of the atoms needs to be much larger than the length of the apparatus such that the atoms in the atomic beam are not deflected on their way from the oven to the glass cell see next section However the stronger constraint is that during an experiment i e on the order of a minute cold atoms should not collide with hot atoms 3 2 1 The Vacuum Chamber A possible way to high loading rates using sodium and lithium is a linear setup with ovens a Zeeman slower and a glass cell Figure 3 1 Overview of our vacuum apparatus see text for more details In the following the numbering relates to figure 3 1 Our setup consists of an arrangement of two ovens for sodium 1 and lithium 2 heated up to 270 C and 350 C respectively They are connected through an angled mixing nozzle 3 that allows liquid sodium to flow back into the reservoir These temperatures are needed in order to increase the very low vapor pressures for these two elements cf 164 65 Sodium and lithium vapors are brought together in the lithium part of the oven and finally diffuse into the apparatus through a conical hole with inner diameter of 4mm referred to as oven nozzle 4 Pressures and thus the ratio of emitted atoms into the ultra high vacuum chamber and the total flux 34 3 3 Zeeman Slower can be adjusted by
26. this slight but sufficient loss on the perpen dicularly polarized component Another change is that the optical axis points out of the plane thus the angle between incident beam and optical axis varies while 21 Chapter 2 Theory Transmission f Af f Af 2 f E ARIZ Figure 2 6 The upper figure shows transmission curves for Lyot filters with arbi trary thicknesses d red dotted line 2d violet dashed line and 4d blue solid line Below the overall transmission of a subsequent arrangement of these three previous filters is shown Af is given by the free spectral range of the thinest plate Af do 2 the FWHM is approximately equal to the FWHM of the thickest plate i e Af 4do 2 rotating the birefringent filter As a consequence n nol can be changed and the transmitted wavelength can be tuned However an important disadvantage of a real birefringent filter is that one does not hit the ratio 2 1 perfectly well As a consequence maxima do not overlap automatically for all wavelengths specified in equation 2 7 but one has to rotate the Lyot filters slightly relative to each other in order to get this close to ideal overlap for the desired wavelength range For a more detailed description refer to 58 In our case the FSR is on the order of several tens of THz 2 1 3 4 Thin and Thick Etalon TNE and TKE The birefringent filter discussed above allows for tuning across the gain width of the dye ye
27. 4 Increasing field and decreasing field slowers do not need a zero crossing However they have their maximal magnetic field close to the MOT or zero field close to the MOT i e Zeeman slower light is resonant with the atoms in the MOT This means that the magnetic field starts at about 600 G and decreases down to 200 G In the zero field domain atoms have to repolarize because of the change in magnetic field direction leading to a change from o to o light in the rest frame of the atoms The advantage of such a configuration is that the magnetic field close to the magneto optical trap is relatively small about 200G So compensation of the magnetic field can be achieved with little effort This is necessary since the magneto optical and magnetic trap would be perturbed On the other hand the Zeeman laser beam is still far detuned relative to the unshifted resonance and MOT operation is not disturbed by this light The only disadvantage is that one needs repumping light since during the repolarization at zero field any magnetic substate of the excited state is populated So decays to the ground state 358 2 F 1 may occur and atoms in that dark state would be lost for further slowing Another point is that sufficient time for this repolarization process needs to be provided resulting in a longer slower 37 Chapter 3 Experimental Setup 3 4 Magnetic Fields For magneto optical trapping magnetic trapping and tuning of scattering
28. 4 3 5 3 Frequency Generation 45 3 5 4 The MOT setup iia sendet aan 47 3 5 4 1 Repumping Light for Sodium and the Dark Spot AAA s ate bende as 47 4 First Measurements 51 4 1 A Provisional Absorption Imaging System 51 4 1 1 Optical Density of an Atomic Cloud and the Beer Lambert ar D ar A al 51 41 2 A Provisional Imaging System 53 vi 4 2 Estimating the Atom Number in the Sodium MOT R sum and Outlook 5 1 Current Progress of the Experiment 5 2 1 Gage gn wid Bede wes Sadia ed Sodium Data Atomic Beam Shutter b l General Aspects er Bl User MAU Glee se eh ee B 2 1 Installation as Sarah B 2 2 Choosing Setpoints and Operation Bo Ehe Circuit as aS eee l Del Programming a Ao oles are B 5 Source Code u aoe Eur ek Bee RE Beam Profiler C 1 Application Notes ci C 1 1 Wannsee 22 Eunctionaliy o roria Oho a een ed C 2 1 Overview Graphs and Fitting C 2 2 Reducing Stripes and Saving Results C 3 Some Comments on the Programming and Fitting Spectroscopy Cell and Doppler Free Laser Locking The Spectroscopy Cell D 2 Lock in Scheme hee Sa oe OE eo we A py RF Drivers for High Frequency Components Danksagung Contents 67 a kd 67 ee ae 67 sm b ti 68 25725 68 68 47777 69 71 71 ae 72 75 77 vil 1 Int
29. 43 Chapter 3 Experimental Setup dalan end UB Bena 1 4MHz G 214G 300MHz 3 3 A Doppler end Vena A 50 MHz 3 4 where ug denotes Bohr s magneton This total detuning of 350M Hz corresponds to 170MHz relative to the laser frequency 3 5 2 5 Zeeman Slower Repumper In regions of strong magnetic fields within the Zeeman slower the distance be tween the cycling transition levels and the other energy levels is increased such that the loss process described in the penultimate section is strongly suppressed Repumping is only necessary in the domain of the zero crossing of the magnetic field In the rest frame of the atoms a change of the magnetic field changes the character of the light from o to o and vice versa thus pumping from one out ermost magnetic substate mp 3 to the other mp 3 is accomplished During this repolarization process decays into the F 1 ground state are no longer forbidden Consequently it is here that the repumper has to act The repumper needs to be detuned by 1713 MHz minus the Doppler shift at the point of the zero crossing However since it is directly modulated onto the Zeeman slowing beam as described in section 3 5 3 this Doppler shift drops out 3 5 2 6 Imaging When doing experiments with cold atoms most of the information on the system is derived from images To take a picture absorption image the atomic cloud is illuminated briefly with a weak collimat
30. Faculty of Physics and Astronomy University of Heidelberg Diploma thesis in Physics submitted by Stefan Weis born in Heilbronn November 2007 Setup of a Laser System for Ultracold Sodium Towards a Degenerate Gas of Ultracold Fermions This diploma thesis has been carried out by Stefan Weis at the Kirchhoff Institute for Physics under the supervision of Prof Dr M K Oberthaler Aufbau eines Natrium Lasersystems zur Erzeugung ultrakalter entarteter Fermigase In dieser Diplomarbeit wird ein neuer Aufbau zur Erzeugung ultrakal ter Natrium und Lithiumgase vorgestellt Ziel dieses Experiments ist die Herstellung entarteter Fermigase aus fermionischen Li Atomen die mittels bosonischer 5 Na Atome sympathetisch gek hlt werden Daf r wurde das Natrium Lasersystem entworfen und installiert Ein wichti ger Schritt war die Implementierung einer magneto optischen Falle f r Natrium In dieser Arbeit soll der bisherige Aufbau beschrieben und eine Einf hrung in die Thematik der ultrakalten entarteten Fermigase gegeben werden Setup of a Laser System for Ultracold Sodium Towards a Degenerate Gas of Ultracold Fermions This thesis presents the first part of a new experimental setup for ul tracold Na and Li gases The aim of this experiment is to achieve Fermi degeneracy within a sample of fermionic Li atoms A laser system for bosonic Na has been designed and set up As a first ex perimental result a magne
31. Profiler m file You ll see the graphical user interface GUI shown in figure C 1 First select the color channel that gives the best results fits your laser color using the radiobuttons Afterwards make sure the peak gray values are neither close to saturated nor too low by either changing the light intensity or choosing an adequate setting of the exposure time and sensitivity in the camera s control panel By default the GUI shows three graphs that are updated about three times per second depending on your computer s performance On the upper left a false color image is shown on the bottom left right a signal proportional to the row color sum of the image is shown as well as a Gaussian fit to the profile This summation has been done in order to smooth the curve that is fitted For a perfect two dimensional Gaussian beam this yields the same results since integrations along the x or y axis are independent providing only a constant factor If you are only interested in a visual examination of the beam the button fitting disables the fitting procedure leading to an increased repetition rate For very noisy images put the slider averaged to a higher value C 2 2 Reducing Stripes and Saving Results In case you are using infrared lasers e g Nd YAG there may vertical stripes appear that perturb the fitting Use the reduce stripes functionality to get rid of them if your beam is not too big in the vertical direction After fi
32. amine 6G R6G and DCM 4 Dicyanomethylene 2 methyl 6 p dimethylaminostyryl 4H pyran respectively In the following we will concentrate on our laser for sodium however the results are generic for nearly all dyes In section 2 1 1 some laser basics needed in the subsequent chapters are given In section 2 1 2 some special features of dyes are discussed Finally in section 2 1 3 we will show how to achieve single mode operation of a dye laser at a desired wavelength introducing the mode selective elements in a dye ring laser 2 1 1 Some Laser Basics This section is not meant to be a profound introduction to laser physics The aim is to recall some laser basics that will be needed in the following For further details refer to any standard text book e g 50 for a general introduction or 51 52 for specific questions on dye lasers The first working laser a ruby laser was built in 1960 by Theodore Maiman Six years later the first dye laser was invented by chance in the group of Fritz P Schafer 53 Examining the saturation characteristics of cyanine the reflectivity of about 4 of a polished cuvette was sufficient to enable lasing Prerequisite for the construction of a laser is an active medium amplifying in coming light coherently i e same phase and wavelength Therefore a population inversion needs to be achieved since only then stimulated emission dominates the absorption in the gain medium This cannot be achieved in t
33. an Clean in this context means that no perturbing interactions like for example in solids exist There are no electron phonon interactions no electrostatic interactions among electrons and with the ionic cores of the lattice However the most important thing is that within a sample of trapped ultracold fermions there are several tunable parameters that do not exist in other systems such as density by means of the restoring force within the trap temperature one can stop cooling at any point and even scattering lengths The latter has the most dramatic consequences and will be discussed in the following 1 2 1 Feshbach Resonances Consider a sample of atoms containing different spin states of fermions A priori collisions between two atoms with different spin states will only happen if they Source http www uibk ac at exphys ultracold atomtraps html 4We will come back to that point in section 1 3 1 2 Degenerate Fermions approach each other to some distance comparable to the diameter of the atoms i e several ag 5 107 m Bohr radius that is much smaller than the inter atomic distance about 100nm in an ultracold gas Thus collisions will happen only rarely This changes drastically if a magnetic field is applied and tuned in the vicinity of a so called Feshbach resonance 15 These resonances lead to scattering lengths a that exceed by far the geometric extensions of the bare atoms In figure 1 2 a plot of the Feshbach
34. and designation of levels in atoms refer to any standard textbook like 50 12Often both of the beams are attributed the same polarization however this is a question of the reference frame 28 2 2 Magneto Optical Trapping Na D transition 223 SP 58 3MHz 34 3MHz Cycling Transition N eilt o Kel be OJA O NIS E 19 m 3 m 2 m 1 m 0 m 1 m 2 m 3 Z S a F 3 F E E El El E E 2 a o F 2 E mp 2 m 1 m O m 1 m 2 2 F F F F F 3 Sis 1 7716GHz a b Figure 2 10 a Magnetic substates of the cycling transition used for sodium b Level scheme of sodium where gm and gm are the Lande factors of the involved states see figure 2 11 64 65 This equation holds only for low magnetic fields as long as the nuclear spin is mainly coupled to the spin orbit momentum of the valence electron compared to the coupling to the external magnetic field We now have to modify equation 2 14 a last time including equation 2 15 Now the scattering rate finally reads 5 2 1 S Yscatter A schematic of the line shifts is given in figure 2 11 Choosing the right magnetic field direction implies a force always directed to the magnetic zero In conclusion there are two effects leading to magneto optical trapping When ever an atom has got a certain velocity its absorption line is shifted into resonance with a beam traveling in opposite direction Whenever an atom is
35. antum field theory a small number of measured quantities like for example the electron s mass and charge are needed However this allows for quantitative predictions of physical observables Whenever such an approach is feasible the corresponding theory is called renormalizable In a next step the known action on a microscopic scale is extended to an effective action on a macroscopic scale Since FR is not based on perturbation theory it can be used for the description of strongly correlated systems The outstanding point about ultracold gases is now that the microscopic behavior of these atoms is very well known in contrast to for example in high energy physics Taking this as a starting point and extending this to a macroscopic scale yields macroscopic observables e g relations between correlation lengths and core sizes of vortices A complementary method is Quantum Monte Carlo simulation See 27 for a very detailed review and 28 for a more recent example on degenerate Fermi gases close to a Feshbach resonance Basically one uses some quantum mechan ical model e g many body Schr dinger equation path integral formalism as a starting point and defines some initial wavefunction Then random walks are used 8This is in some way comparable to the UV catastrophe in Rayleigh Jean s law where a discrete description needs to be applied for high photon momenta 13 Why Lithium AND Sodium for solving path integrals or for consecutiv
36. apter 1 Introduction versa Using this many phenomena can be described in Bose Einstein condensed systems If needed higher order interactions can be integrated into this model 1 2 2 2 Strongly Coupled Fermions A more interesting point and a currently very active field of research is the descrip tion of strongly coupled fermionic systems Evidently an approach similar to the GPE can not exist since fermions have to occupy orthogonal quantum mechanical states with an overall antisymmetric wave function As long as they are weakly interacting perturbation theoretical approaches still hold However the most ex citing physics takes place right on and next to Feshbach resonances where the systems are strongly interacting Directly on a Feshbach resonance the scattering length diverges and systems are supposed to show a unitary behavior i e they show the same characteristics on all length scales in quark gluon plasma but also in neutron stars One method currently developed at the Theoretical Institute of the University of Heidelberg 124 25 is a quantum field theoretical approach called Functional Renormalization FR In quantum field theory 26 one typically encounters divergences for high momenta i e small distances These arise from a breakdown of the continuum description of fields However this problem can be solved by introducing a so called ultraviolet cutoff and replacing higher momentum physics by measured quantities For qu
37. ays into the wrong F 1 ground state are dipole forbidden Since the hyper fine splitting of the excited state is small compared to the linewidth excitations to the F 2 state actually may occur Electrons may now decay into the F 1 state and are no longer trapped Putting a repumping beam resonant with the F 1 to P 1 1679 1859 transition returns atoms into the cycle Taking the relevant Clebsch Gordan coefficients into account the F 1 to F 2 transition would have been favorable The advantage of our selected resonance is that given a repumping beam that counterpropagates the Zeeman slowed atomic beam under an angle of about 50 the atoms experience shifted light that is resonant with the F 1 to F 2 transition In conclusion the F 1 to F 1 transition is used for repumping of trapped atoms the F 1 to F 2 transition for the incident beam that is about to be trapped Yet the effect of this method still has to be evaluated 3 5 2 4 Zeeman Slower The magnetic end field and the desired speed of the atoms define the detuning for the Zeeman slowing beam We arbitrarily chose the magnetic field maximum close to the glass cell to be 214G with a speed of the atoms of Vena 30 m s cor responding to the capture velocity of the MOT The contribution for the relevant Zeeman shift and Doppler shift reads now 13Sidebands are directly modulated onto the Zeeman slower beam HHyper fine levels of excited state are not resolved
38. by the number of electrons protons and neutrons in its nucleus and shell each contributing spin 1 2 Thus for odd numbers like in Li 3 3 3 atoms are fermionic and for even numbers like in Na 11 11 12 they show bosonic behavior Second in The transition to the superfluid phase of Heor type I superconductivity are Bose Einstein condensed systems but strong interactions between particles complicate these systems heavily Strong interactions result in a reduced fraction of condensed atoms about 10 in superfluid 4He 1 1 Quantum Statistics Bosons and Fermions a regime in which quantum properties play a role particles do no longer behave like classical point like particles but also show wave like properties This was first postulated in 1924 by L V de Broglie who attributed a wavelength h Na 1 4 1 4 to a particle of momentum p 7 These properties are only relevant if the in ner structure of the particle is small compared to its wavelength since otherwise particle particle interactions are mainly arising from the interaction of electronic shells A second point is that the inter atomic distance needs to be on the order of this wavelength Or in other words 1 5 where n is the number density In statistical quantum mechanics this product is also called phase space density or degeneracy parameter This already gives a rough estimate for temperatures needed in a system of given number density
39. called Bose Einstein condensate BEC and has been predicted as early as in 1924 by Satyendranath Bose and Albert Einstein 1 2 3 In 1995 finally the first Chapter 1 Introduction pure BECs have been realized experimentally by Eric Cornell and Carl Wieman at JILA 4 and Wolfgang Ketterle at MIT 5 Bose Einstein Condensate Classical Gas Na YO Degenerate Fermi Gas E Mions 6 Decrease Temperature Li Increase Density Figure 1 1 Cooling down a gas leads to the appearance of quantum properties See text for more details Fermions on the other hand must not occupy one single quantum mechanical state This is manifest in the Fermi Dirac distribution the analog to the afore mentioned Bose Einstein statistics 1 N s ri 71 1 3 This has first been claimed by W Pauli in 1925 known as Pauli s principle Ul timately the fermions will reduce the system s energy when cooled down by occu pying the lowest empty states available This leads to a sharp transition between occupied and empty states at a certain energy level known as the Fermi energy This phase is referred to as a degenerate Fermi gas There are two more essential principles for an understanding of quantum gases First in 1940 Wolfgang Pauli could show that the spin of a particle determines its quantum properties 6 Bosons carry integer spin whereas fermions have half integer spin The behavior of atoms as a whole is determined
40. ccur In a last step electrons in energy level 4 relax to the ground state 1 Taking into account the absorption of the pump beam stim ulated emission and spontaneous emission as well as all non radiative processes one can establish rate equations allowing to calculate the lasing threshold i e the pump power needed for lasing and the time dependent behavior Figure 2 1 Term scheme of rhodamine 6G S denotes electronic singlet states T the corresponding triplet states 3Evidently 2 and 4 may be any state within the upper and lower band 14 2 1 Dye Lasers A last important property of dye lasers is that in steady state operation dye lasers with electrons occupying only one single excited state all electrons in 3 will run in a single longitudinal mode at every time though fast mode hops may occur leading to an effective multi mode operation This is due to what is called homogeneous line broadening of the gain medium Electrons in 3 serve as a reservoir for all possible longitudinal modes As a consequence only one mode at a time will be amplified at the expense of all the others Stimulated emission within the gain medium amplifies the longitudinal mode with wavelength A and intensity 1 according to ana x N3 I A where N3 denotes the density of electrons in the 3 state Hence the most intense mode depopulates electrons in the 3 state the most and prevails the others that gain less gradually and get damped o
41. d W Ketterle Two Species Mixture of Quantum Degenerate Bose and Fermi Gases Physical Review Letters 88 160401 2002 W Demtr der Experimentalphysik 3 Springer Verlag GmbH 2005 F P Sch fer Dye Lasers Springer Verlag GmbH 1977 C V Shank Physics of Dye Lasers Rev Mod Phys 47 3 649 Jul 1975 F P Sch fer W Schmidt and J Volze Organic Dye Solution Laser Applied Physics Letters 9 306 oct 1966 Thieme Chemistry R MPP Online Version 3 0 Georg Thieme Verlag KG 2007 Fuh R A Du H Li J Corkan A and Lindsey J S PhotochemCAD A Computer Aided Design and Research Tool in Photochemistry Photochem istry and Photobiology 68 141 1998 S Rabien Wirtsgalarien von Quasaren und Der Laserleitstern f r das Very Large Telescope PhD thesis LMU Miinchen 2004 B Lyot Optical Apparatus with Wide Field Using Interference of Polarized Light C R Acad Sci 197 1593 1933 S M Kobtsev and N A Sventsitskaya Application of Birefringent Filters in Continuous Wave Tunable Lasers A Review Optics and Spectroscopy 73 114 jul 1992 83 Bibliography 59 H J Metcalf and P van der Straten Laser Cooling and Trapping Springer Verlag New York Inc 1999 60 A Ashkin Acceleration and Trapping of Particles by Radiation Pressure Phys Rev Lett 24 4 156 Jan 1970 61 S Chu J E Bjorkholm A Ashkin and A Cable Experimental Observation of Optically Trapped Atoms Phys Rev Lett 57 3
42. d for the magneto optical trap On the optical table the MOT light is coupled into a single mode polarization maintaining fiber that is split into three pairs of fibers by a micro optics device manufactured by Canadian Instruments While the power in two fibers of one pair is very close to equal 2 different pairs vary by up to 10 This fiber splitting unit greatly reduces the optics for the MOT we only need to collimate the light and choose the right polarization Therefore a mount holding fiber holder A 4 plate and a 100 mm lens has been built see figure 3 6 Figure 3 7 Dark spot 3 5 4 1 Repumping Light for Sodium and the Dark Spot MOT The repumping light is coupled into a multimode fiber mainly because the high frequency AOM destroys the Gaussian mode needed for efficient fiber coupling and no purified Gaussian mode is needed for repumping In a further step we will implement a so called dark spot MOT As described in section 2 2 6 the number of 18 Actually there are geometries requiring only four beams 83 however achieving big stable MOTs like this is difficult 19 Actually a more power saving strategy is to retroreflect the MOT beams and to change the polarization using a A 4 plate in front of the mirror However in an optically dense MOT the returning beam is attenuated and balancing might be a problem In spite of this the group of van der Straten realized the biggest sodium MOT ever in this configuration 48
43. day rotator Faraday active crystal combined with a strong permanent magnet the original polarization is restored for a beam traveling in the desired direction but it is turned for a coun terpropagating beam which then suffers losses on subsequent circulations As discussed in the previous section the mode with the highest gain per circulation will prevail inhibiting counterpropagating beams 5A modified ELS MonoDisk laser 19 Chapter 2 Theory 2 1 3 2 Selecting a Longitudinal Mode Single mode lasers that can be tuned close to an atomic resonance are a prerequisite for laser cooling Their frequency accuracy needs to be smaller than the natural atomic linewidth so for example 1 MHz in the case of sodium line width T 2r 10MHz This corresponds to a relative frequency accuracy of about 10 Simultaneously the tuning range of the gain medium several tens of nanometers for R6G should be conserved It is intuitively clear that this is not feasible with only one optical element as there has to be a tradeoff between the free spectral range FSR the frequency distance between two transmitted modes the width of the transmission peak and the tunability Instead several hierarchic filters are inserted namely a birefringent filter a thin and a thick etalon and finally the resonator itself that can be tuned by a tweeter and a galvo plate These elements provide wavelength selectivity on scales of several nanometers i e THz to
44. dreds of uK 63 2 2 3 Magneto Optical Trapping of Multilevel Atoms There are several changes when switching to real atoms First of all there are evidently more relevant and accessible energy levels like can be found in the simplified level scheme of sodium in figure 2 10 where F denotes the total angular momentum including the electron s and core s spin and the angular momentum of the electrons Each of the F states is now composed of 2F 1 degenerate magnetic substates as shown in figure 2 10 for two energy levels Even though all this looks quite differently from what has been discussed the cooling mechanism described above still works even better see 2 2 4 The laser is tuned slightly below the resonance F 2 to F 3 and pairs of o and o polarized counterpropagating beams are used This drives the atoms to either of the outermost substates shown in figure 2 10 depending on the direction the atom is moving in and results in a Doppler cooling scheme Up to now atoms may be cold but trapping is not yet achieved since no position dependent force is established Inserting two magnetic coils in an anti Helmholtz configuration yields a magnetic quadrupole field The magnetic field introduces a position dependent Zeeman shift of the magnetic sublevels The resonance frequency of the cycling transition is shifted by 1 A Zeeman A mr id Imp uBB T 1 4 MHz G B x 2 15 For an introduction to selection rules
45. drops out since this integration has already been done implicitely when projecting the cloud onto the camera On a discrete grid defined by the CCD pixels the integral turns into a sum N Y 4 4 8 Nx Ny where m ny denote the pixels of the whole image and A is the area of one pixel VThis equation remains valid if there is some optical magnification since the scattered light stays the same 52 4 1 A Provisional Absorption Imaging System 4 1 2 A Provisional Imaging System A provisional absorption imaging system has been set up using a PCO pixelfly vga digital CCD camera system with a resolution of 12 bit The dimension of one pixel is 9 9 x 9 9 um An imaging beam resonant to the atomic transition with very low intensity 10 uW cm compared to the saturation intensity has been installed It passes through a single mode polarization maintaining fiber and is collimated on the vacuum table Its angle of incidence onto the glass cell is about 20 This reduces interference fringes due to multiple reflections within the glass cell optical density 0 45 vertical position um 0 05 1 15 2 25 3 35 4 45 horizontal position um Figure 4 1 Optical density of the magneto optical trap For a quantitative analysis of the atom number three images with atomic cloud without atomic cloud and without imaging beam have to be taken The latter dark image is subtracted from the others in order to compensate
46. e case of rho damine 6G For linear molecules the eigenenergies of the n th eigenstate then reads h n 12 where h is Planck s constant m is the electronic mass and L is the length of the box potential For ring like structures the same equation holds however there are two eigenstates to each eigenenergy there are no fixed boundaries resulting 2 1 15 Chapter 2 Theory Pinacyanol IS 2 7 27 2 1 Rhodamine 6G C2H5 C2H5 Cu Phtalocyanin m US Figure 2 2 Left Structure of two generic dyes with a linear and ring like shape 51 Right Structure of hodamine 6G 54 in distinct sine and cosine like solutions Every state can now be occupied by two electrons Thus N z electrons occupy the lowest N 2 states The lowest absorption band arises from transitions from then N 2 to the n N 2 1 state The corresponding energy difference and wavelength is h N 1 Emin 2 2 8mL 2 2 8me D max 2 A h N 1 2 3 where c denotes the speed of light In R6G rough estimations of the absorption wavelength are no longer that easy as it is neither linear nor circular but there are several connected circles of 7 electrons However for R6G molecules the qualita tive behavior described above still holds In figure 2 2 one can find the chemical structure of R6G Its absorption o A and fluorescence P A spectrum is depicted in figure 2 3 oz A is the cross section for stimulated em
47. e of Li is about 7 whereas the rest is Li we use enriched lithium containing about 95 Li The natural abundance of 4 K is only about 0 012 and is mainly won in nuclear power stations 11 Chapter 1 Introduction In chapter 2 we will give an introduction to dye lasers to an extent that seemed to be necessary to understand the specific properties also problems of our laser system Another point we would like to touch upon in this chapter is some theory on magneto optical trapping Chapter 3 will give an overview on what has been set up in the first year of this experiment We will describe the vacuum system the Zeeman slower and the magnetic coils briefly before giving a more extensive description of the laser system that has mainly been developed and set up under the author s responsibility during the first months In part 4 we will present first measurements of the properties of our magneto optical trap for sodium atoms The appendix finally contains some sodium data and several important exper imental tools realized during this diploma thesis namely a beam profiler based on a webcam a microcontroller based atomic shutter driver for our vacuum appa ratus the spectroscopy cell for Doppler free saturated spectroscopy of sodium and the driver electronics for high frequency modulators 12 2 Theory 2 1 Dye Lasers We use a laser system based on two Radiant Dyes Dye Ring Lasers For sodium and lithium we chose rhod
48. e relies on reproducible results on a timescale of hours thus any heating effects have to be minimized The assembly that is currently set up consists of four coils of the type shown in figure 3 2 two above and two below the glass cell with 15 windings each Hollow squared wire has been used such that the heat can be removed by temperature regulated water flowing through The assembly is embedded in epoxy 3 4 2 Magnetic Trap For magnetic trapping we will use a quadrupole magnetic field Therefore the multi function coils are switched to an anti Helmholtz configuration by means of a bridge circuit based on insulated gate bipolar transistors IGBT for more details refer to the diploma thesis of Anton Piccardo Selg 75 The magnetic field is zero in the center of the assembly and increases linearly in radial and axial direction with a gradient that is twice the value for the radial direction For magnetic trapping the atoms need to be transferred to some low field seeking state that is immune towards spin exchange collisions These atoms will now oscillate around the zero There are only two states for sodium with a non vanishing magnetic moment satisfying these conditions namely F mr 1 1 and 2 2 The 1 1 state is preferred by far for creating sodium BECs yet it undergoes spin exchange collisions with lithium see discussion in 71 48 So the 2 2 state is the one to choose Optical pumping in a small magnetica
49. e steps in phase space 1 2 3 Current Research Topics A Short Summary In this section we would like to give a short overview on current research topics in the leading groups of the international community This list does not claim to be complete e In the group of Deborah Jin at JILA p wave Feshbach resonances are examined 19 e The groups of Randy Hulet Rice University 29 and Wolfgang Ketterle MIT 30 deal with imbalanced spin mixtures and explore phase diagrams in such systems e At Duke University the group of John Thomas investigates thermodynam ics at a Feshbach resonance 31 e The group of Rudi Grimm University of Innsbruck is doing spectroscopy 32 on ultra cold degenerate Fermi gases and examines their dynamics 33 e In the group of Christophe Salomon ENS the transition from a gaseous to a crystalline phase is investigated 34 Recently expansion experiments have been done 35 1 3 Why Lithium AND Sodium We have chosen fermionic Li and bosonic Na for our experiment But why a bosonic part if all this is about fermions Even though there is in fact interesting physics 136 37 when dealing with a mixture of a degenerate Fermi gas and a BEC this mainly has to do with our cooling strategy In a first step laser cooling is done Yet to reach temperatures below the critical temperatures this is not sufficient since only temperatures on the order of hundreds of uK can be achieved for lithium In BEC physics
50. echnology 2003 84 Bibliography 72 M Repp Aufbau einer Vakuumapparatur f r Experimente mit Ultrakalten Fermionischen und Bosonischen Quantengasen Master s thesis Kirchhoff Institute for Physics University of Heidelberg 2007 73 J Krieger Title Undisclosed Master s thesis Kirchhoff Institute for Physics University of Heidelberg 2008 74 Witte A and Kisters T and Riehle F and Helmcke J Laser cooling and deflection of a calcium atomic beam J Opt Soc Am B 9 7 1030 July 1992 75 A Piccardo Selg Title Undisclosed Master s thesis Kirchhoff Institute for Physics University of Heidelberg 2008 76 Z Hadzibabic S Gupta C A Stan C H Schunck M W Zwierlein K Dieck mann and W Ketterle Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms Physical Review Letters 91 16 160401 Octo ber 2003 77 M O Mewes M R Andrews N J van Druten D M Kurn D S Durfee and W Ketterle Bose Einstein Condensation in a Tightly Confining dc Magnetic Trap Phys Rev Lett 77 3 416 Jul 1996 78 D E Pritchard Cooling Neutral Atoms in a Magnetic Trap for Precision Spectroscopy Physical Review Letters 51 1336 October 1983 1791 W Petrich M H Anderson J R Ensher and E A Cornell Stable Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms Phys Rev Lett 74 17 3352 Apr 1995 80 K B Davis M O Mewes M R Andrews N J van Drut
51. ed beam of resonant light 0MHz 180MHz and scatters photons proportional to its local particle density This produces a shadow of the cloud that is recorded by a camera 3 5 2 7 Transfer into a Magnetic Trap As briefly discussed in section 3 4 2 the atoms need to be transferred to the low field seeking 2 2 state before being transferred to the magnetic trap Efficient transfer can be achieved by shining in o polarized light resonant to the F 2 to F 2 transition after switching off the MOT This transfers the atoms to the dark 16The Zeeman slower beam is according to its working principle on resonance all along the magnetic field thus especially at the zero field Consequently a repumping beam shifted by 1720MHz is resonant at that point too 44 3 5 The Laser System F mp 2 2 state without any further heating Therefore light at 58 MHz 4 122MHz is needed FSR 1 5 GHz Fabry Perot Transmission arbitrary units sfs 0 1 2 400 600 800 1000 1200 1400 1600 1800 2000 2200 Time arbitrary units Figure 3 5 Transmission through a scanning Fabry Perot interferometer with free spectral range FSR of 1500 MHz of a beam behind an EOM driven at 1720 MHz Since the FSR is inferior to the mode spacing the signal of the n th sideband is folded back by a multiple of the FSR and appears at n times 220 MHz from the peak The incoupled RF power was about 2 5 W 3 5 3 Frequency Ge
52. en D S Durfee D M Kurn and W Ketterle Bose Einstein Condensation in a Gas of Sodium Atoms Phys Rev Lett 75 22 3969 Nov 1995 81 J P Denschlag Kalte Atome in singul ren Potentialen PhD thesis University of Innsbruck 1998 82 E A Donley T P Heavner F Levi M O Tataw and S R Jefferts Double Pass Acousto Optic Modulator System Review of Scientific Instruments 76 6 063112 2005 83 F Shimizu K Shimizu and H Takuma Four Beam Laser Trap of Neutral Atoms Optics Letters 16 339 mar 1991 85 Bibliography 84 C G Townsend N H Edwards C J Cooper K P Zetie C J Foot A M Steane P Szriftgiser H Perrin and J Dalibard Phase Space Density in the Magneto Optical Trap Phys Rev A 52 2 1423 Aug 1995 85 T Ottenstein A New Objective for High Resolution Imaging of Bose Einstein Condensates Master s thesis University of Heidelberg 2006 86 Erkl rung Ich versichere dass ich diese Arbeit selbst ndig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe Heidelberg den 20 11 2007 Unterschrift
53. en observables or maybe the distance and radius of vortices in a superfluid system As discussed before bosons in a BEC occupy the same quantum mechani cal state Thus a simple and widely used approach is to reduce the N particle Schr dinger equation to an effective one particle one in a mean field approach 2 SWFA A Veal EOP 17 Here Ves denotes some external potential N is the particle number and g is the coupling constant that is proportional to the scattering length thus positive for repulsive and negative for attractive interactions Mean field in this context means that one assumes the other atoms to be homogeneously distributed in space cre ating a net background field corresponding to the last term of the Hamiltonian This equation was first derived by Gross 21 22 and Pitaevskii 23 in 1961 and is called Gross Pitaevskii equation GPE The non linear term in the Hamil tonian assumes the interaction of the particles to be point like Only two body s wave scattering processes are taken into account This is justified if the temper ature is low enough and higher order scattering freezes out and if the inter atomic distance is big compared to the scattering length such that three body collisions do not occur For attractive interaction the non linear term will decrease the total energy of the system and thus lead to an increased particle density n and vice Note that n x W F t where n is the particle density Ch
54. epopulate the upper laser level Moreover absorption of laser light to Ta is actively damping the laser beam within the cavity Concluding the most important principles of Dye Lasers are that to a good approximation the energy levels 2 are empty whereas 3 is populated by means of the pumping light However there are actually collision induced losses to the triplet state absorbing lasing light thus at a certain point higher pumping powers do not yield any higher output powers but saturate Furthermore the dye liquid may heat up locally leading to instable laser operation Both effects can be reduced using a dye jet such that dye transferred to the triplet state is quickly removed out of the laser As a rule of thumb the higher the speed of the jet the higher the pumping power can be chosen 2 1 3 Singlemode Operation of Dye Lasers In this section the dye laser we use will be described In figure 2 4 one can find a schematic overview of our ring dye laser manufactured by Radiant Dyes Laser amp Accessories GmbH All elements are mounted onto an invar bar for minimal thermal expansion of the cavity Central feature is a polished nozzle of optical quality ejecting a thin film of dye solution in the following referred to as dye jet through the cavity into a catcher hose The pressure needed is provided by a Radiant Dyes dye circulator pushing the dye solution through the dye nozzle at up to 7 bar 4For our system saturation starts at
55. eracting only with the light field but they also reabsorb photons that have been scattered by the others These incoherent processes introduce a repulsive force between the atoms and keep them apart This peak density can be increased by nearly a factor of ten if the atoms in the center of the trap are in the untrapped F 1 ground state This can be achieved by removing the repumping light in the center of the trap dark spot 69 Without a repumping beam the atoms fall down to the dark F 1 state after about 100 scattering processes and radiation trapping is reduced Repumping only occurs at the edge of the MOT keeping the atoms in the middle 13Originally SPOT is a short form for SPontaneous force Optical Trap 31 Chapter 2 Theory 32 3 Experimental Setup The following chapter will give an overview of our setup with an emphasis on the laser system We use two optical tables of 3m x 1 5m each for the vacuum setup on the one hand and for all the optics and light preparation on the other hand The two are interconnected using optical fibers for the light and a BNC bus for the exchange of data 3 1 Introduction In figure 3 1 we show an overview of our vacuum system that has been set up during the last year As it is characterized in more detail in Marc Repp s diploma thesis we will only give a short summary on design guidelines and functionalities in the following The objective of our apparatus is to cool as many Li atoms as
56. evaporative cooling 38 is done Thereby the fastest atoms are removed Efficient cooling is only achieved if collision induced rethermalization occurs sufficiently fast These collisions are mainly s wave scattering processes since higher order collisions freeze out at the given temperatures During this step the atoms are typically trapped in a magnetic trap thus the sample is usually There are groups with all optical setups 39 In this case atoms are transferred from the MOT to an optical dipole trap directly and this problem does not arise But for this enormous laser powers typically several tens of watts are needed for achieving sufficiently high potentials Chapter 1 Introduction spin polarized The important point about spin polarized fermions is now that s wave collisions are forbidden by Pauli s principle for low temperatures while higher order collisions are freezing out Consequently thermalization would slow down drastically for decreasing temperatures As collisions between different spin states are still allowed at low temperatures cooling down to degeneracy can be done by using different spin states In fact this has been the first working solution ever and was chosen by the group of Deborah Jin at JILA Disadvantageous is that one loses about 99 of the atoms This can be circumvented using a second approach called sympathetic cooling 40 It is based on creating a conventional BEC of bosons and during this
57. evel driver available for this camera the win video interface of the operating system had to be used This provides only some basic configurations concerning trigger modes and resolution The winvideo inter face acquires its data from the camera driver The aforementioned control panel influences parameters like exposure time sensitivity or any gamma corrections Unfortunately gray values are no longer proportional to the intensity but the slope flattens for higher gray values Available video devices can be found using gt gt imaghwinfo Acquiring images from any video source can be obtained using the MAT 69 Chapter C Beam Profiler LAB videoinput functionality e g gt gt vid videoinput winvideo 1 RGB24 640x480 Available trigger settings can be found using gt gt set vid For manual triggers one obtains the acquired image sequence using gt gt trigger vid gt gt color_image getdata vid 1 For more details refer to the MATLAB Help topic Basic Image Acquisition Procedure The fitting procedure uses a slightly modified version any warnings have been suppressed of the fminsearch function where starting values are es timated based on the maximal and minimal values found within the smoothed line profiles together with their position plus a roughly estimated half width Within the source code you can find a more detailed description 70 D Spectroscopy Cell and Doppler Free Lase
58. ffects like for example sub Doppler cooling arise always in view of the specific situation in a sample of sodium and lithium atoms Finally the working principle of a magneto optical trap MOT will be described 2 2 1 Light Forces on Two Level Atoms There are two distinct effects exerting forces on atoms They are briefly described in the following Note that only a classical motivation is given A quantum mechanical derivation can be found in the book of Metcalf and van der Straten 59 The way to go is to establish a Hamiltonian containing a term coupling the eigenstates by means of a dipole operator to an electro magnetical field Diagonalization leads to the appearance of new eigenstates called dressed states Inserting these states into a statistical density matrix approach and intro ducing spontaneous emission leads to the optical Bloch equations describing the time dependent behavior of these systems 25 Chapter 2 Theory 2 2 1 1 Dipole Force Here this aspect is only given for the sake of completeness However an important point for our future experiment will be the construction of an optical dipole trap 60 61 precisely based on the optical dipole force We would like to give a classical justification for this force In a classical ap proach a two level atom can be compared to a damped electrical resonator with the resonance frequency wo and a dipole momentum of p driven by an inhomoge neous alternating electrical fie
59. gains for each component VCO Since these two items are the most expensive objects on our optics table special care has been taken that the maximal specified RF powers of AW for the BOM and 1 W for the AOM cannot be exceeded under any circumstances The output power has been measured using a Tektronix TDS 7254 oscilloscope with a 50Q input impedance Control Control Power Voltage 5V 12V Voltage Monitor T Fixed Voltage Power vco Attenuator Variable Attenuator Amplifier Output Minicircuits Minicircuits Minicircuits Kuhne electronic ZX95 1900V BW S1W2 ZX73 2500 KU PA BB233 BBA max Output Power for AOM 7 3dBm 3dB 3 3dB 35dB 33dB 34dBm 2 6W for EOM 6 7dBm 7dB 3 3dB 35dB 33dB 29dBm 790mW Figure E 1 Schematic of the RF drivers used for the high frequency AOM and EOM Below a summary of the levels attenuations and gains for each component is listed The power amplifiers feature a power monitor output This voltage Vp is related to the RF pover Prr as can be seen in figure E 2 A quadratic fit valid Voltage Controlled Oscillator Chapter E RF Drivers for High Frequency Components for the frequency range between 1700MHz and 1900MHz yields P ao VE Vp ag E 1 where ao 0 48 W V a 0 18 W V and az 0 02 W The driver electronics is connected to the crystals by means of a Minicircuits CBL 6FT SMSM coaxial cable The VCO control voltage needs to be sufficiently stable compa
60. glass cell For a more detailed description of our setup and an analysis of vacuum con ductivities and pumping speeds refer to Marc Repp s diploma thesis 72 3 3 Zeeman Slower A Zeeman slower has been installed in order to increase the flux of slow capturable atoms in the glass cell Loading of a magneto optical trap requires slow atoms with speeds of less than 30m s as a rough estimate whereas the velocity distribution of sodium and lithium atoms is centered around 800 m s and 1600m s for the given oven temperatures Evidently only a very small fraction is captured without further precautions In the Zeeman slower atoms can be decelerated to the required speed Our slower is designed such that sodium and lithium atoms with initial 2When heating up the vacuum the different specific expansion coefficients of steel and glass would in either case make the cell burst So the expansion coefficient needs to be changed gradually 35 Chapter 3 Experimental Setup speeds of up to 700m s can be slowed down The most probable velocity of the atoms leaving the oven is about the same for sodium but about twice as high for lithium Thus the ratio of slowed sodium atoms is much bigger than that of lithium Since more sodium than lithium is needed for the further cooling this is not a restricting point Below I will briefly describe the functional principle of a Zeeman slower and some design criteria of the one we set up For more details on an
61. gned such that together Numerical integration of the absorption along the axis The non linearity of the absorptional cross section has been taken into account 72 D 2 Lock in Scheme with a 30nF capacitor in a series LC circuit amplitudes of up to B 6G at the resonance frequency of about 80 kHz can be obtained using a standard Voltcraft type 7202 wobbel function generator See figure D 4 for an explanation This modulation is detected on the photodiode and fed into a lock in amplifier giving rise to an output signal proportional to the derivative of the Doppler free spec trum cf figure D 4 for a schematic The measured sodium spectrum is shown in figures D 5 and D 6 o E F Absorption Signal A U level shift Derivative A U Frequency A U Figure D 4 Working principle of the lock in scheme The upper graph shows a schematic of a resonance line The latter is shifted up and downwards by means of a RF magnetic field In the frame of the atoms the laser frequency is modulated relative to their resonance frequency leading to a modulated absorption signal Feeding this signal into a lock in amplifier yields an output signal proportional to the modulation amplitude i e the derivative of the line Locking on top of the line can now be accomplished by feeding this signal in this framework also referred to as error signal back into a control circuit Windings
62. he transition and T the natural linewidth 6 is the laser detuning 6 w wo 2 2 1 2 Light Pressure Force Near and at resonance also dissipative processes can be used for cooling Whenever an atom absorbs light it gathers a photon s momentum Pphot hk Once in the excited state there needs to be some kind of deexcitation through spontaneous or stimulated emission before a second excitation process can start Consider now the two possible processes Absorption followed by stimulated emission does not change the atom s momentum and will not serve for cooling since incoming and outgoing photons are the same Spontaneous emission on the other hand leads to scattering of photons to random directions thus there is a mean net momentum of NPpro acquired after scattering N photons This force is called light pressure force However since electrons in the excited state have a non zero lifetime 7 1 T the rate of scattered photons is limited It can be calculated using the following equation 59 T 5 2 21494 28 5 1 1 is called saturation parameter with the transition and polarization spe cific saturation intensity J and 6 denotes the detuning of the laser relative to the resonance In the limit of high saturation Yscatter approaches T 2 This corresponds to the situation that half of the atoms are in the excited state and spontaneous decays happen at a rate of T Even though the recoil of the atom when absorbing one photon
63. hermal equilibrium or in a system of only two levels since Boltzmann s factor cannot exceed 1 In ap propriate systems of three and four levels this becomes possible A key feature of lsee http www pat2pdf org patents pat3353115 pdf for the original patent Strictly spoken this holds only on timescales bigger than the natural linewidth 13 Chapter 2 Theory these media is that the laser transition s lifetime constituted of all non coherent decay channels spontaneous emission non radiative decay is long compared to all other decay processes Let us have a look directly at the level scheme of rhodamine 6G in picture 2 1 that will be discussed in some more detail in the next section Refer to figure 2 1 for notations of the various lifetimes and states Lifetimes and 74 are on the order of picoseconds or even sub picoseconds see next section for a brief justification and very small compared to 73 0 1 us Thus one can assume in a simple approach that all electrons excited to 2 decay into 3 instan taneously In a first step pumping light excites electrons from the ground state 1 to any sublevel of the excited state 2 Non radiative transitions induced by non elastic collisions populate the state 3 There are now several decay paths The most desirable is stimulated emission to some sublevel of the ground state 4 Furthermore spontaneous emission to the ground state or transitions to the triplet states may o
64. hor Stefan Weis stefan weis gmx de Wee ee Ve ee Ae ee processor PIC 12F683 clock frequency 1 MHz internal aaa SK SR SR a a SK SR R EEEE EEEE SR SR SR R K SR SR SR R K SR SR SR R K SR SR SR SK SR SR 21 SK SR SR R R K SR SR SR ak a A ak SR ak ak A ak ak Ak Include register names for the PIC 12F683 include lt P12f683 INC gt we we Configuration power up timer no watchdog internal oscillator masterclear enabled __CONFIG _MCLRE_ON amp _PWRTE_ON amp _WDT_OFF amp _INTOSCIO ak ak 3k 3K ak 9K 3K 3K 3k 9k 3K 3K ae SR SR 2K 3K SR I SR SR SR I A SR SR SR SK SR SR SR SK SR SR SR SK SR SR a A aK a ak k A ak k k define variables Flags equ 0x20 LSB is set to present status of TTL Input temp equ 0x21 for calculations k akk ak ak ak aka SK SK K ak SR SR SR aa 3K 3K I II I I I II SR SR SR 3K aI I I I I SK SK SR SSR SR SR SR aa OR a a ak ak ak ak a K aaa K K K ak org 0x00 goto Main org 0x04 Interrupt Routine it might have been more elegant using interrupts on change of Pin GP5 but this direct implementation can be debugged more easily 3 3 aka a ae ae R SR R 2k SK SSR SR SR SR SR R SR SR SR ae a a S SR SR SR SR SRR SR SR SR ae a SR R ae ak SK SR SK SR SR SR SR ae fe SK ok R K SR SR ak ak ak ak ak Initialization 64 B 5 Source Code k k k gt K ak ak ak k k ak k ak ak ak ak ak ak ak ak ak
65. iency is as high as 15 however the outcoupling aperture is too small Since we need to recycle the zeroth order only 5 can be achieved without clipping at the housing This is still sufficient for our purposes For the Zeeman slower repumping light an electro optical modulator New Focus Inc with a center frequency of 1720 MHz is used An incoupled RF field is reinforced resonantly inducing a field dependent optical path by means of the Pockels effect This sine shaped modulation creates sidebands to a laser beam passing through at nf where n is an integer and f is the radio frequeney17 In figure 3 5 there is an example for what the spectrum looks like behind the EOM The drivers for these high frequency modulators have been assembled during this thesis and are briefly described in chapter E clean air x s r 1 7 7 ris aperture Eu du 4 1 Single mode fiber Figure 3 6 a Acousto optic modulator at 1 9GHz The housing has been man ufactured in order to keep dust apart b MOT beam assembly The Power within the n th sideband is proportional to BAGIN where is proportional to the square root of the RF Power and J denotes the n th Bessel function 46 3 5 Laser System 3 5 4 The MOT setup As mentioned in section 2 2 a MOT requires three pairs of beams with opposite circular polarizations and a quadrupole field The multi functional coils described in section 3 4 are also use
66. ignal of less than 1Hz The servo will start to switch between the two positions given by the potentiometers Adjust them for complete blocking passage of the atomic beam Finally connect the TTL input to the computer control B 3 The Circuit The electronics must provide the following features e two analog inputs corresponding to the user selectable setpoints on and off e one TTL input switching between the two states e one PWM output to servo one status output The easiest way how to achieve this is by means of a small microcontroller We chose a PIC 12F683 manufactured by Microchip with 8 pins and with 8 bitted data memory It is equipped with 6 multifunction I O pins configurable as analog inputs or digital I O Especially it contains an internal clock oscillator and a PWM module In figure B 1 one can find the circuit now exhibiting a minimum of additional parts namely two potentiometers voltage dividers R2 and R3 one voltage regu lator for protection purposes of the microcontroller only IC1 and a light emitting diode as status indicator As a reference voltage of the internal 10 bit analog to digital converter the supply voltage is used so setpoints actually do not depend on any fluctuations of the latter B 4 Programming The programming part has been done in Assembler Implementation was done in a straightforward way yielding as few debugging as possible This section is lFor more details refer to
67. ission In a liquid solution large molecules experience in general more than 101 colli sions with solvent molecules per second This means that the system reequilibrates 16 2 1 Dye Lasers 10715 cm onu 500 600 700 Wavelength nm Figure 2 3 Fluorescence and absorption os A spectrum of a rhodamine 6G solution 1075mol 1 The data is based on measurements of Fuh et al 1998 55 Knowing the lifetime of the excited state one can calculate the cross section for stimulated emission 0em A for more details cf 56 An approximate curve for triplet state absorption has been inserted on the order of picoseconds at room temperature ending up in the vibronic ground state of the first excited electronic state or in any higher state according to Boltz mann s distribution The emission spectrum can now be obtained by mirroring the absorption spec trum around the frequency of the purely electronic transition Absorption excites electrons from the electronic and vibrational ground state to any excited electronic and vibrational state emission starts in the electronic excited state and vibrational ground state to any vibrational state of the electronic ground state see figure 2 1 Up to now we did not take triplet states into account For every electronic excited singlet state there is a corresponding triplet state One can show with a simple argument that triplet state energies are inferior to the corresp
68. ition and thus pushes out the atoms This influence can only be reduced by choosing a relatively small detuning Up to now we are using 15MHz 165MHz as a compromise of the values used in the groups of W Ketterle 20MHz and P van der Straten 11MHz since no systematics have been done yet Hany bold frequencies given are from here on meant to be relative to the cycling transition bold italic frequencies relative to the laser frequency 12One might think of choosing a MOT frequency below the lowest hyper fine state at lt 110MHz however in this case only Doppler cooling will occur At the same time repumping power needs to be increased drastically since transitions to the F 1 state are no longer forbidden an inconvenient point as will be seen in the next section 42 3 5 Laser System 23Na STi rel Intensity Orel MHz MHz MHz MHz MOT 15 165 ol 25 157 MOT repumper 1679 1859 4 mW 228 410 Zeeman slower 350 170 Iz gt Isa 344 162 Zeeman slower repumper 1713 o 1096 12 116 66 Imaging 0 180 lt 1mW 0 182 Transfer 58 122 Loa r Lock Frequeney 180 0 0 5 mVV 182 0 Table 3 1 Frequencies needed where 6 is the detuning relative to the atomic resonance and is the detuning relative to the laser frequency 3 5 2 3 MOT Repumper A priori the cycling transition in sodium is closed where closed means that dec
69. ive would have been Toptica s TA modules based on tapered amplifiers offering up to 500 mW But even though dye lasers are more demanding than solid state lasers they offer a mode quality that is far better than laser diodes Additionally the output power is sufficiently high 3 5 2 Frequencies All frequencies had to be derived from one single laser using acousto optical AOM and electro optical EOM modulators Consequently the design of the laser sys tem has to comply with some basic requirements These diodes have rated powers of up to 120 mVV However their typical wavelength is 662nm and they need to be heated up to about 65 C for 671 nm 40 3 5 Laser System Na D transition 223 Li D transition 58 3MHz 3 P 2 3 2 3 Pas 4 4MHz o 2 8 a N N E 5 EIT Ola 2 ci E N co E o 3 la l gt de al 00 o o Clo o Ee Q N wv gt LO Y de s o 2 2 3S 1 7716GHz 355 Figure 3 3 Level scheme of 23 Figure 3 4 Level scheme of Li e efficient use of laser power e maximal tunability of frequencies e cost effective use of standard optics thus especially use of standard AOMs Figures 3 3 and 3 4 show the level schemes of the relevant transitions in Na and SLi For simplicity the D lines have been omitted They are about 400 GHz and 10 GHz below the corresponding D lines for sodium and lithium and thus not interferi
70. l bias field allows to transfer the atoms to the 2 2 state see section 3 5 2 7 Once in the center there is no longer a polarizing field spin flips occur and atoms in untrapped states are lost In order to avoid this several trap con figurations have been developed such as clover leaf 77 Ioffe Pritchard 78 or time orbiting potential TOP 79 traps We use a standard quadrupole magnetic field combined with a blue detuned laser beam focused into the center of the trap optical plug 80 The light shift potential induced by this beam keeps the trapped atoms away from the magnetic zero Transistors designed for switching of high currents i e several hundreds of Amp res 5 Collisions that change the internal spin state of the atoms transferring them from a trapped to an untrapped state 6Not only the 2 2 state but also the 12 1 and 2 0 are at least weakly trappable in a the magnetic trap Since there are spin exchange collisions between them the latter need to be completely removed in order to close this loss channel 71 48 76 TVVe will use some of the pump laser power at 515 nm 8Potential attributed to the optical dipole force of the light see 2 2 1 1 39 Chapter 3 Experimental Setup 3 5 The Laser System As described in section 2 1 3 our laser system is based on two Radiant Dyes Ring Dye Lasers They are pumped by a frequency doubled Yb YAG disc laser with two outputs at gt 10 W each Fo
71. lag bit is also equal to zero return to loop no change choose A D Channel wait for about 5 microseconds i e 5cycles until multiplexing has been finished set Flag bit to current TTL level 0 set Status indicator LED to high off else run Changed function 35 cycles finished Start conversion Is conversion done No test again 8 MSBs of conversion contained in ADRESH rescaling to duty cycle range of Servo will be done in the following for 5 to 10 duty cycle rounding to 6 significant bits will be effectuated in the following i e discard the two least significant bits rotation to the right twice yields 64 1024 6 25 range adding 45 0x2D yields a range of 45 1024 4 4 to 99 1024 9 7 rotating right rotating right clear two most significant bits add Siles 45 two least significant bits of result have to be written into CCP1CON lt 4 5 gt swapped and modified bits in temp are s s state written into CCP1CON without changing configuration bits Eliminate two LSBs of result that have already been treated clear two most significant bits sand put result into CCPR L C Beam Profiler During this diploma thesis software for a custom made beam profiler has been developed Electronics and the CCD chip have been taken from a Logitech Quick Cam Pro 4000 webcam and put into a housing made of brass Thereby lenses and infrared
72. ld E Z t En cos wt Solving the differential equation for a driven damped oscillator yield a phase difference between field and oscillator of ME gt 2 10 where 3 is the damping coefficient This equation is plotted in figure 2 9 red detuning blue detuning ob 004 Frequency Figure 2 9 Relative phase between a driven oscillator and the driving field as a function of detuning In electrodynamics the time averaged potential energy U of a dipole in an electric field is given by 62 U cos d P 2 2 11 The qualitative behavior of U as a function of detuning can be deduced from figure 2 9 For blue detuning cos is positive thus the mean potential is increased whereas red detuning lowers the particle s potential energy Finally the amp dependence of the electrical field induces a dipole force Fa VU 6 2 12 Concluding a dipole in an inhomogeneous alternating field is torn into the maximal field for red detuned w wo negative light but seeks low fields for blue detuned 8Other names are reactive force gradient force and redistribution force 26 2 2 Magneto Optical Trapping light w wo positive For the conditions met in the aforementioned dipole trap a quantum mechanical calculation dressed state approach gives the following approximation F a SIE 2 13 ae where 2 denotes the laser beam intensity the saturation intensity of t
73. lithium laser system still needs to be set up Yet the optics needed 97 Chapter 5 R sum and Outlook is already available and this part should not take too much time In parallel a proper imaging system needs to be set up Probably similar to the one installed in our rubidium BEC experiment next door 85 A final thing is the transfer of the species to an optical dipole trap and this is where the actual experiment will begin 98 A Sodium Data A simplified level scheme of sodium can be found in figure 3 3 The vapor pressure is plotted in figure D 2 Table A 1 finally lists some important optical properties of sodium For further information refer to 64 Frequency without Hyperfine Shift 27 508 848 716 2 13 THz Transition Energy hwo 2 104 428 981 77 eV Wavelength Vacuum A 589 158 3264 15 nm Lifetime T 16 249 19 ns Natural Line Width FWHM r 27 9 9795 11 MHz Recoil Velocity Ur 2 9461 cm s Recoil Temperature Wp 2 3998 uK Doppler Temperature Tp 235 uK Saturation Intensity cycl transition Isat 6 2600 51 mW cm Table A 1 Sodium D Transition Data 64 99 Chapter A Sodium Data 60 B Atomic Beam Shutter B 1 General Aspects During this work a driver for the atomic beam shutter has been realized About 10cm behind the oven nozzle there is a rotary feedthrough inserted into our vacuum system that is connected to a thin plate made of stainless s
74. marco and D S Jin Onset of Fermi Degeneracy in a Trapped Atomic Gas Science 285 5434 1703 September 1999 15 H Feshbach Unified Theory of Nuclear Reactions Reviews of Modern Physics 36 1076 October 1964 16 C Stan Experiments with Interacting Bose and Fermi Gases PhD thesis Massachusetts Institute of Technology 2005 17 B Marcelis E G M van Kempen E G M Verhaar and S J J M F Kokkel mans Feshbach Resonances with Large Background Scattering Length In terplay with Open Channel Resonances Physical Review A 70 1 012701 2004 18 I Bloch J Dalibard and W Zwerger Many Body Physics with Ultracold Gases ArXiv e prints 704 April 2007 19 J P Gaebler J T Stewart J L Bohn and D S Jin P Wave Feshbach Molecules Phys Rev Lett 98 200403 2007 20 M J Holman and P A Wiegert Long Term Stability of Planets in Binary Systems Astronomical Journal 117 621 jan 1999 21 E P Gross Hydrodynamics of a Superfluid Condensate Journal of Mathe matical Physics 4 195 February 1963 22 E P Gross Structure of a quantized vortex in boson systems Nuovo Cimento 20 3 454 1961 80 Bibliography 23 L P Pitaevskii Vortex Lines in an Imperfect Bose Gas Sov Phys JETP 13 1961 24 S Diehl The BCS BEC Crossover in Ultracold Fermion Gases PhD thesis University of Heidelberg 2006 25 S Fl rchinger Renormierungsgruppe von Quantenfeldtheorie bis Quanten statistik Exp
75. nd transparency diminishes after a while In this case windows can be cleared by covering them with some aluminum foil and heating up the oven to temperatures close to 200 C for half an hour The sodium mirror will disperse increasing the transmission of the windows The oven can be heated up with a solenoid put onto the aluminum cylinder not shown in figure D 1 The windings are two at a time in opposite direction avoiding magnetic fields Magnetic fields in the center of the oven are well below 0 5G for 2A of heating current A temperature of about 130 C has shown to be optimal for spectroscopy representing a tradeoff between the size of the signal and optical transmission where 130 C correspond to about 1 9 A 18 V electrical power in steady state In figure D 3 a theoretical calculation shows the transmission of an incident beam for several saturation parameters S 107 3 5 6 Qa 2 10 E 5 5 5 2 108 8 9 5 3 10 E gt E 2 y Nu 1072 110 120 130 140 150 160 170 180 20 40 60 80 100 120 140 160 180 FH Temperature C Temperature C Figure D 2 Vapor Pressure of Figure D 3 Transmitted intensity sodium as a function of tempera as a function of temperature for ture 64 different saturation values D 2 Lock in Scheme Directly onto the glass cell RF coils for a high frequency modulation of the mag netic Zeeman sublevels have been wound They are desi
76. neration All but two frequencies are shifted using acousto optical modulators AOM fabri cated by Crystal Technology Inc These contain a radio frequency RF transducer attached to a transparent crystal of tellurium dioxide Driving the transducer with RF powers of typically 0 5 W creates a running sound wave inside the crystal Pho tons crossing the crystal will experience Raman processes i e absorb or emit a phonon if the angle of incidence is such that quasi momenta and energy is con served The photon s frequency can change in that way by integer multiples of the RF where only the first orders are used in our experiment Passing the light twice through one single AOM by retroreflecting results in a shift of twice the RF Commercial standard AOMs are available typically at frequencies of 80 110 and 200 MHz with bandwidths of about 10 Our optics schemes have consequently been designed such that all high power beams are shifted by AOMs close to res onance in order to achieve highest possible efficiencies The AOMs used in a double pass configuration have been set up similar to the proposition in 82 Two exceptions concern the repumping light For the MOT repumper we use an AOM at about 1 9GHz provided by Brimrose Corporation of America see figure 3 6 This device is far off from being standard and to our knowledge 45 Chapter 3 Experimental Setup can only be fabricated by the aforementioned company Nominal effic
77. ng with any process discussed in the following In the following we will give an overview of the frequencies needed The lithium laser system has not been set up until now so only a proposed scheme is given in figure 3 9 and table 3 1 3 5 2 1 Locking the Laser to an Atomic Resonance Both lasers are locked internally to a temperature stabilized reference cavity with a short term stability of about 1MHz However on the timescale of hours or even seconds a stability well below the natural linewidth of lithium Tz 27 6 MHz and sodium T ya 27 10 MHz can not be guaranteed Long term stability is provided by locking the laser to an atomic transition At room temperature a cell filled with some Na or Li is not sufficient to show appreciable absorption due to 10The time constant of the feedback loop is limited by the mechanical 1 galvo plate within the reference cavity thus on the order of hundredths of seconds 41 Chapter 3 Experimental Setup the minuscule vapor pressure of both species Therefore absorption cells have to be heated up to about 130 C and 240 C respectively A sodium spectroscopy oven has been set up during this diploma thesis based on an evacuated glass cell filled with sodium More technical details on this and on Doppler free spectroscopy can be found in appendix D For lithium a somewhat more elaborate scheme is needed because lithium diffuses into glass and chemically reacts with it This lead
78. of SLi with bosonic Na as a refrigerant During this year we have installed a laser system for cooling and trapping of Na and a vacuum apparatus Also a new lab has been equipped from zero see page iii A first magneto optical trap for sodium atoms with on the order of 10 atoms has been seen however these results have to be considered as preliminary In the meantime the final coils together with working offset coils for compen sation of any stray magnetic fields have been installed and the vacuum apparatus formerly slightly bent has been straightened Furthermore a running version of the computer control has just been released Altogether this should allow for increasing the number of trapped atoms drastically Especially since from now on systematic optimization will be greatly simplified 5 2 Outlook An important milestone on our way to degeneracy will be the first BEC of sodium Before there are still some steps to take First of all our multi function coils installed since end of October 2007 need to be put into operation Afterwards the first thing to do will be to increase the atom number within the magneto optical trap using a dark spot technique and the transfer into the magnetic trap Another ongoing project is the design and construction of the microwave antenna for 1 8GHz and drivers that will be used for RF evaporative cooling Once the BEC has been obtained the lithium part will be tackled For this purpose the
79. onding singlet state energies The overall wave function for a system of two electrons in states m and n at positions ri ra and with spins s s needs to fulfill Pauli s principle i e the total wave function needs to be antisymmetric toward particle exchange There are now two different possibilities Either spins are parallel or antiparallel to each other leading to a symmetric or antisymmetric spin function As a consequence the corresponding wave functions needs to be antisymmetric or 17 Chapter 2 Theory symmetric respectively Ym nhri T in Ti T symmetric wave function 2 4 Was dmnlr r2 nm Ti 72 antisymmetric wave function 2 5 Symmetry now leads to electrons being closer to each other than in the asymmetric case leading to a higher potential energy Thus triplet state energies are always lower than the corresponding singlet states and transitions occur mediated by collisions with the solvent Figure 2 1 shows the relevant level scheme of rhodamine 6G Optical pumping with green light e g 515nm or 532nm excites electrons from the Sp ground state to a substate of S By means of non radiant processes collisions with solvent molecules electrons occupy the lowest S state 3 with 1 2 3 and 4 forming a four level system There are several loss processes intrinsic to the gain medium First of all spontaneous emission from 3 to 4 and collisions inducing transitions to T d
80. os for a PhD Thesis 2007 26 M E Peskin and D V Schroeder An Introduction to Quantum Field Theory ABP 1995 27 W von der Linden A Quantum Monte Carlo Approach to Many Body Physics Physics Reports 220 53 nov 1992 28 G E Astrakharchik J Boronat J Casulleras and S Giorgini Equation of State of a Fermi Gas in the BEC BCS Crossover A Quantum Monte Carlo Study Physical Review Letters 93 20 200404 November 2004 29 G B Partridge W Li Y A Liao and R G Hulet Pairing Phase Separation and Deformation in the BEC BCS Crossover Journal of Low Temperature Physics 148 323 August 2007 30 Y I Shin C H Schunck A Schirotzek and W Ketterle Phase Diagram of a Two Component Fermi Gas with Resonant Interactions 2007 31 J E Thomas J Kinast and A Turlapov editors Universal Thermodynamics of a Strongly Interacting Fermi Gas 2006 32 M Mark F Ferlaino S Knoop J G Danzl T Kraemer C Chin H C Nagerl and R Grimm Spectroscopy of Ultracold Trapped Cesium Feshbach Molecules Physical Review A 76 4 042514 2007 33 M J Wright S Riedl A Altmeyer C Kohstall E R Sanchez Guajardo J Hecker Denschlag and R Grimm Finite Temperature Collective Dynam ics of a Fermi Gas in the BEC BCS Crossover Physical Review Letters 99 15 150403 2007 34 D S Petrov G E Astrakharchik D J Papoular C Salomon and G V Shlyapnikov Crystalline Phase of Strongly Interacting Fermi
81. possible on the order of 107 should be realistic 70 to the lowest temperatures in reasonably short time Limiting factor for the number of fermions in the degenerate phase is the number of sodium atoms initially trapped since sympathetic cooling results in huge losses of sodium atoms The group of W Ketterle reports on losses of about 50 for lithium during the transfer to the magnetic trap and another factor of two during the sympathetic cooling stage 71 Thus a lithium atom number on the order of 10 in the MOT should be sufficient These 107 Li atoms need to be cooled at the expense of sodium So cooling will only be possible as long as sufficiently many sodium atoms are available Taking into account transfer efficiencies from the MOT to the magnetic trap of about 30 for sodium less than 1 of the sodium atoms are left after evaporative cooling where this number highly depends on the starting and final temperature as well as the atom number in the fermionic cloud As a rule of thumb sympathetic cooling is efficient as long as there is at least a comparable number of sodium in TUp to 70 can be achieved with more elaborate transfer mechanisms to the F mp 1 1 state 48 However these are not applicable in our case See discussion in 71 and in section 3 4 2 33 Chapter 3 Experimental Setup the magnetic trap 71 Thus an initial 1017 sodium atoms should be captured in the MOT 3 2 Vacuum System For
82. profiles are shown Evidently the optical density is low Using equation 4 8 yields an atom number on the order of 2 107 A Gaussian plot to the line profiles yields diameters FWHM of d 1 20 2 mm in vertical axial and d 1 92 2 mm in horizontal radial direction Evidently the cloud has not a Gaussian shape in horizontal direction but the row profile is slightly tilted Our interim coils could not be switched fast enough d 4 2 Estimating the Atom Number in the Sodium MOT This may arise as soon as the effective potential of light and magnetic forces is no longer harmonic but contains higher order terms Probably the Zeeman slower beam contained some close to resonant stray light pushing the magneto optical trap slightly Assuming the cloud to have the same thickness in both radial directions the peak density in the center is given by 3 Nmazr 10 atoms cm 4 9 The low optical density indicates that multiple scattering processes do not yet play a dominant role In this case atoms move in a quasi harmonic potential 72 84 and one would expect a Gaussian density distribution This justifies the Gaussian fit in figure 4 2 Refer to Marc Repp s 72 diploma thesis for loading and loss rates as well as temperature estimates 99 Chapter 4 First Measurements 56 5 R sum and Outlook 5 1 Current Progress of the Experiment One year ago we started to set up an experiment on degenerate Fermi gases
83. r Locking Locking the sodium dye laser is done by means of a Doppler free saturated ab sorption spectroscopy Therefore resonant light passes a sodium vapor cell twice The pump beam saturates an atomic transition for atoms of a certain velocity v whereas the counterpropagating probe beam excites atoms of velocity v Evi dently both beams do not interact as long as v 52 0 For v 0 a so called lamb dip appears showing the Doppler free profile of the line See figure 3 8 for the optics scheme used for locking D 1 The Spectroscopy Cell Figure D 1 Schematic drawing of the sodium spectroscopy cell a end caps reducing the solid angle b takes up the glass cell c evacuated glass cell filled with sodium d viewport might also serve as a cold spot not examined in detail e RF coils feedlines not shown f aperture for included thermo couple type K not shown not shown heating coils The vapor pressure cf figure D 2 for sodium needs to be increased in order to 71 Chapter D Spectroscopy Cell and Doppler Free Laser Locking obtain a sufficient absorption signal Therefore a small oven has been developed and built during this thesis A drawing can be found in figure D 1 It takes a glass cell filled with sodium Thorlabs CP25075 NA The solid angle of the aperture has been minimized in order to avoid heat sinking effects at the windows leading to steaming up with sodium Yet this cannot be completely suppressed a
84. r p wave scattering of two atoms in the same spin state 19 gt This system can be treated similarly to an atom in a light field using a dressed state approach 17 See also 18 for a nice introduction 6 Bardeen Cooper and Schrieffer 1 2 Degenerate Fermions 1 2 2 Theoretical Approaches 1 2 2 1 BEC Theory The inner structure electronic configuration magnetic and electrostatic proper ties of species usually used in cold atom experiments is well known Using this knowledge collisions involving two atoms can be described theoretically In cold atom experiments with atomic velocities on the order of millimeters per second the collisional energies are weak compared to the binding energies of the innermost shells Thus only valence electrons contribute Two particle interaction featur ing scattering and formation of molecules of atoms is well understood and the Schr dinger equation can be solved numerically However moving to higher atom numbers quickly exceeds any available com puter powers Furthermore already three classical particles e g two planets in a solar system 20 may behave chaotically and solving such problems exactly is impossible On the other hand the exact solution of the Schr dinger equation of say 10 cold atoms cannot be examined in an experiment anyway One is rather interested in macroscopic quantities that can be determined in an experiment like for example densities temperature correlations betwe
85. r sodium we are using a dye solution of rhodamine 6G in ethylene glycol The concentration is set such that the pump laser beam is absorbed to about 98 within the dye jet yielding concentrations of about 0 8 g l The maximal power exceeds 1W for 8W of pumping power However stable mode hop free operation is currently only possible at about 800 mW This is probably due to heating of the dye in the center of the pumping spot With a higher dye jet speed higher stable powers should be easily achievable Therefore either more powerful dye circulators or a thicker dye nozzle are needed The dye used for Lithium light is DCM with a concentration of about 0 8 g l in a mixture of 50 ethylene glycol 18 benzyl alcohol and 32 propylene carbonate Its power exceeds 1 5 W Stability tests have not yet been performed 3 5 1 Why Dye Lasers Because there is currently no well developed alternative for sodium light at 589nm At the Laboratoire Kastler Brossel ENS Fabrice Gerbier et al are currently developing a solid state laser at 589nm using frequency mixing of diode lasers However power output is below 400mW up to now Already available commer cially are small Diode Pumped Solid State DPSS modules with a second harmonic generator but they are far away from single mode In the case of lithium there are actually cheap laser diodes with powers of up to 100 mW available e g laser diode type HL6545MGS that are used in DVD recorders Another alternat
86. red to the natural linewidth of sodium For a stability to about 1MHz the control voltage must not vary by more than 15mV Therefore a tunable voltage reference has been set up using a A723 precision voltage regulator according to the application information Basic High Voltage Regulator in the datasheet 3 5 3 L S 25 s 1 5 on 3 1 72GHz 957 e 1 90GHz fit to both 0 5 i i i 0 0 5 1 0 1 5 2 0 2 5 Monitor Out Voltage V Figure E 2 RF power as a function of the monitor out voltage http focus ti com lit ds symlink ua723 pdf 76 F Danksagung An dieser Stelle m chte ich mich bei allen Personen bedanken die zum Gelingen dieser Arbeit beigetragen haben Mein spezieller Dank gilt dabei Prof Markus K Oberthaler fiir die Aufnahme in seine Arbeitsgruppe und die M glichkeit dieses neue faszinierende Experiment mit aufzubauen Nicht nur sein Enthusiasmus f r Physik die vielen wertvollen Diskussionen und die langen N chte im Labor mit ihm sondern auch zahlreiche Abende au erhalb des Labors machten dieses Jahr zu einem sch nen und sehr wertvollen Jahr Prof Annemarie Pucci f r die Begutachtung dieser Arbeit Peter Kr ger dem wissenschaftlichen Leiter dieses Experiments den Pionie ren dieses Experiments Marc Repp Jan Krieger und Jens Appmeier sowie Elisabeth Brama und Anton Piccardo Selg die k rzlich zu uns gesto en sind Danke f r die vielen sch
87. resonances for a pair of 914 atoms of opposite spin is given There are two known Feshbach resonances at relatively low magnetic fields a very narrow one at 543 G and a very broad one at 837 G The latter will be used in our experiment since it demands only little accuracy when ramping magnetic fields Moreover one can even choose whether the particles effectively attract a lt 0 or repel a gt 0 each other 6000 4000 2000 0 1 400 800 1200 2000 Scattering Length a 4000 6000 Magnetic Field Gauss Figure 1 2 Scattering length a in units of the Bohr radius ao for Li atoms with opposite spin as a function of the magnetic field 16 There are two Feshbach resonances at 543 G not resolved in this plot and 837 G and a zero crossing of the scattering length at 528G for the two lowest hyper fine states in high fields For a gt 0 the interaction is repulsive otherwise attractive We cannot give a detailed introduction to Feshbach resonances in this work so let us just motivate where they are arising from For a nice introduction refer to 17 In figure 1 3 potential curves for two molecular states of different total angular momentum e g lowest two hyper fine states in high field as a function of the inter atomic distance are plotted The dashed line corresponds to the kinetic energy of the unbound particles in the center of mass system The offset arises from Chapter 1 Introduction
88. rld that have accumulated extended knowledge about laser and evaporative cooling interactions in ultracold samples and BECs in a wealth of different ge ometries and potentials BECs have been put into double wells 8 9 lattices 10 effective two dimensional structures 11 have been rotated 12 Yet there is still a lot of exciting physics to be done Common to them all is that these systems can be well described theoretically as described briefly in section 1 2 2 One research area that has developed recently and that we would like to join is the creation of degenerate Fermi gases of neutral atoms DG The first such DG gas has been observed in the group of Deborah Jin at JILA in Boulder in 1999 13 14 Since then several groups have caught up A short overview on current research topics and involved groups will be given in section 1 2 3 There are now various reasons making research on DG so exciting Ultracold degenerate Fermi gases are a model system for nearly any strongly correlated fermionic system Quarks in a quark gluon plasma electrons in solids or neutron stars may serve as examples These systems have in common that many aspects have not yet been captured theoretically and only now theorists are developing methods that are able to treat many body systems of fermions since perturbation theories break down in this case refer to section 1 2 2 for some remarks on this An important advantage of such a system is that it is cle
89. roduction 1 1 Quantum Statistics Bosons and Fermions Every particle elementary or composite can be attributed to one of two groups It can either be a boson or a fermion Let us take a look at figure 1 1 On the left hand side a schematic of a gas at say room temperature in a harmonic trap is shown In quantum mechanics the eigenstates of such a trap can now be calculated yielding equally spaced energy levels for a one dimensional model indicated by the horizontal lines For a classical gas the available states are sparsely populated according to the Boltzmann distribution given by 1 N 7777 1 1 where N denotes the number of particles within a sample in the i th state with an eigenenergy of E at temperature T in a sample of N 3 N atoms Z is some normalization constant in statistical mechanics partition function and kz is Boltzmann s constant Cooling this sample down to very low temperatures while at the same time increasing the density will at some point lead to the appearance of quantum properties For bosons the probability for the common occupation of one sin gle state is increased compared to classical particles satisfying the Bose Einstein distribution given in equation 1 2 where u denotes the chemical potential 1 Ni ETL 1 2 This enhancement leads to a macroscopic occupation of the ground state for high phase space densities i e low temperatures paired with high densities This phase is
90. s to an increased opacity and permanent damage Typically this problem is overcome using a so called heatpipe a relatively long and thin tube made of high grade steel that is heated in the middle The inner surface is covered with a high grade steel mesh The windows attached on either side are kept at room temperature and a small amount of buffer gas is brought into it avoiding any direct collisions with the windows Lithium atoms hitting the walls off the center stick to them and diffuse back to the center along the mesh See 16 81 for a more detailed analysis and concrete implementations The sodium laser is locked to a frequency of 180MHz relative to the cycling transition mainly because this simplifies the whole optics scheme Another point about this is that any stray light is far detuned about 8 relative to the next atomic resonances thus not interfering at any point The laser is not locked directly to the F 2 to F 3 transition but to the most significant feature of the spectrum i e the cross over of the transitions F 2 to F 3 F 2 that is about 29MHz red detuned Therefore one needs to shift up the laser s frequency by about 151MHz before doing spectroscopy A measured spectrum can be found in figures D 5 and D 6 3 5 2 2 MOT For magneto optical trapping light is usually detuned by several linewidths For sodium atoms however a red detuned beam is at the same time blue detuned for the F 2 to F 2 trans
91. sing a sample of particles of density n is attenuated due to scattering processes according to the Beer Lambert law if the scattering cross section o is independent of the light intensity I In exp 4 4 1 where Jp and J denote the incident and transmitted intensity respectively and d is the thickness of the sample The scattering cross section for atoms in a light field is given by 48 00 4 2 1 5 2 h T where oo L 4 3 51 Chapter 4 First Measurements Here is the detuning of the light T is the linewidth T 27 9 98 MHz for D line of sodium and S I Isa is the saturation parameter with the light intensity J and the saturation intensity sat for the given transition op is called on resonance cross section For atoms the Beer Lambert equation holds if 1 1562 Thus if the light intensity is small compared to the saturation intensity Beyond the scattering cross section decreases as a function of 7 For resonant light of low intensity only oo has to be known The optical density being defined as OD In 1 1 4 5 can now be related to the particle density using equations 4 5 and 4 1 OD 4 6 Given an absorption image thus the optical density as a function of two spatial directions x and y z being the direction of the imaging beam the total particle number N can be calculated using 00 00 D N a il gt dx dy 4 7 where the z direction d just
92. t a frequency of approximately 2kHz yielding a Inside the cavity there is another galvo plate that can shift transmission fringes by more than 15GHz 24 2 2 Magneto Optical Trapping frequency modulation of its transmission curve and an amplitude modulation of the output power of the laser The internal lock in amplifier generates the error signal out of the power signal and feeds it back onto the TKE Galvo plate and tweeter are locked to the Reference cavity Frequency locking is now done on any positive slope of the transmission spectrum This is why the voltage level of the reference signal can be shifted to negative values see figure 2 8 for a schematic The zero crossings define lock points It is now sufficient to take the registered and shifted signal as input for the control loops of galvo plate and tweeter Frequency scanning is simply achieved by scanning the reference cavity A feed forward signal is put onto the thin etalon that is followed by the TKE Galvo plate and tweeter cancel any non zero reference cavity signal and consequently follow the sweep 2 2 Magneto Optical Trapping In this section the concept of magneto optical trapping will be introduced briefly After a short introduction to light forces on two level atoms we want to provide a basic understanding of how light forces can be used for cooling of neutral atoms In the last part this concept will be extended to the case of multilevel atoms where new e
93. t it is not sufficiently selective to achieve single mode operation Now we would like to briefly present the function of the next levels in hierarchy two Fabry Perot etalons called thin and thick etalon The thin etalon is a 0 5mm thick glass plate at close to normal incidence with coated surfaces for a reflectivity of about R 20 yielding a FSR of about 200 GHz and a finesse of F 5 arcsin 52 1 4 It can be tuned by slightly rotating Ratio between free spectral range Af and FWHM of the transmission peaks f nomencla 22 2 1 Dye Lasers its mount that incorporates a galvanometer simultaneously This leads to a slight change of the optical path between the two surfaces and shifts the transmitted wavelengths The thick etalon is composed of two adjacent prisms cf figure 2 7 with a small distance between them one of which is mounted onto a cylindrical piezo Wavelength tuning can now be done by changing the voltage applied to the piezo The inner surfaces are cut under Brewster s angle The total thickness of the system is about 10mm resulting in a FSR of about 10 GHz also with a finesse of about 1 4 10mm cylindrical piezo Figure 2 7 Schematic drawing of the thick etalon refer to the text for details 2 1 3 5 Tuning Resonator Modes Galvo Brewster Plate and Tweeter GP and M2 The mode spacing of the cavity is given by Af c l 200 MHz where c denotes the speed of light and the resonator length of abo
94. tan K Dieckmann S Gupta M W Zwierlein A G r litz and W Ketterle Two Species Mixture of Quantum Degenerate Bose and Fermi Gases Phys Rev Lett 88 16 160401 Apr 2002 G Roati F Riboli G Modugno and M Inguscio Fermi bose quantum degenerate k rb mixture with attractive interaction Phys Rev Lett 89 15 150403 Sep 2002 T Fukuhara Y Takasu M Kumakura and Y Takahashi Degenerate Fermi Gases of Ytterbium Physical Review Letters 98 3 030401 2007 J J Zirbel K K Ni S Ospelkaus J P D Incao C E Wieman J Ye and D S Jin Collisional Stability of Fermionic Feshbach Molecules 2007 46 47 48 49 50 51 52 53 54 55 56 57 58 Bibliography M Bartenstein A Altmeyer S Riedl R Geursen S Jochim C Chin J Hecker R Grimm A Simoni E Tiesinga C J Williams and P S Juli enne Precise Determination of 6 Li Cold Collision Parameters by Radio Frequency Spectroscopy on Weakly Bound Molecules Physical Review Let ters 94 103201 2005 C A Regal M Greiner and D S Jin Lifetime of Molecule Atom Mixtures Near a Feshbach Resonance in 40K 2003 K M R van der Stam E D van Ooijen R Meppelink J M Vogels and P van der Straten Large atom number Bose Einstein condensate of sodium Review of Scientific Instruments 78 3102 January 2007 Z Hadzibabic C A Stan K Dieckmann S Gupta M W Zwierlein A Gor litz an
95. teel Depending on the position the atomic beam is blocked by it or not The problem specifications were a sufficiently high shutter speed shutting in much less than 1s without striking the plate against the wall heavily The solution was to use a BLUEBIRD high speed servo model BMS 661 MG HS typically used in model aircraft or cars At a 6 V supply it is specified to be faster than 0 1s for a 60 turn with no load Maximal torque is 50 Nem more than sufficient for the small moment of inertia of the feedthrough plus the plate Servos contain an electric motor a gear and a position sensor Choosing a position is done by sending a pulse width modulated PWM signal to the servo continuously where 5 and 10 duty cycle correspond to the extreme positions Frequency should typically be in the range of 50 to 100Hz An internal control circuit drives the servo to the selected position B 2 User Manual B 2 1 Installation Turn the servo arm such that the screws that are transmitting torque are accessible from behind slightly screw very few turns it to the rotary feedthrough and bring the shutter to a centered position Afterwards clamp the shutter assembly to the vacuum apparatus Make sure it is centered well 61 Chapter B Atomic Beam Shutter B 2 2 Choosing Setpoints and Operation Connect the driver box to the shutter assembly to power supplies for the electronics 5 415V and servo 4 6V up to 1A and an alternating TTL s
96. the data sheet http uvi microchip com domloads en DeviceDoc 41211D_ pdf 2A nice introduction to PIC microcontrollers can be found here http www sprut de electronic pic index htm 62 B 4 Programming Figure B 1 Circuit of the atomic beam shutter driver supposed to give a rough overview for a more detailed documentation refer to the source code B 5 In the Init part in and outputs and their operational mode as well as the A D converters and the PWM module are configured the clock frequency is set to 1MHz Refer to the source code header for further information on the pin assignment The PWM module is connected to the internal Timer 2 module Prescalers allow to reduce the PWM frequency to about 62 Hz The Main function mainly contains an endless loop polling the TTL input pin for changes Whenever TTL levels change the LED is switched the active A D port is toggled triggered and one acquisition is accomplished In a next step the result is divided by 16 i e the result is shifted to the right four times and incremented by 45 This value is written into the register governing the duty cycle Together with a resolution of 10 bit of the PWM module this yields settable duty cycles between 45 1024 4 4 and 64 45 1024 10 6 Since the supply voltage is also used as the reference for the A D converter the acquired value actually does not depend on any voltage changes Finally the controller jumps back into the loop 63
97. the different hyper fine states in the molecule Different magnetic moments of the two spin states allow for shifting the upper potential curve relative to the lower one Preparing the atoms in the open channel the lower curve i e in an unbound state yields a coupling to the closed channel upper curve by means of spin exchange collisions of the nuclear spin If ever the kinetic energy is close to an energy level of the closed channel the eigenstates repel each other avoided transitions leading to a drastically increased scattering length Thereby a lowering increase in energy is equivalent to an attractive a repulsive interaction potential of molecular state potential for two free atoms inter atomic distance Figure 1 3 Schematic of potential curves for two molecular states of different total angular momentum as a function of the inter atomic distance dashed line kinetic energy of the involved particles in the center of mass system At sufficiently low temperature this leads either to Cooper pairing for negative scattering lengths or molecule formation for positive scattering lengths This situation is often referred to as BEC BCS cross over since on the one side of the resonance molecules are condensed into a molecular BEC described by BEC theory on the other side bosonic Cooper pairs described in the BCS theory of superconductors For completeness we mention that there are also Feshbach resonances fo
98. the order of MHZ in the order mentioned above The product of the transmission profiles results in a sharply peaked curve permitting to suppress all but one mode These elements will be described briefly in the following Their assembly is sketched in figure 2 4 2 1 3 3 Birefringent Filter BR The birefringent filter is composed of a three staged Lyot filter 57 with a thickness ratio between two subsequent filters of two The easiest case of a Lyot filter consists of a birefringent plano parallel plate of thickness d the optical axis lying inside the plane followed by a polarizer A laser beam hitting the surface perpendicularly with a linear polarization angled 45 to the optical axis may pass without any polarization changes if the condition In m ll d NAN 2 6 Ne 1 f d N Ir n lld 2 7 is fulfilled Here N is an integer n and m denote the extraordinary and ordinary refractive index respectively thus l n lld is the difference between the optical path lengths within the crystal If this difference is equal to a multiple of the wavelength NAy no phase shift will happen Light with a frequency satisfying equation 2 7 that has been derived using f c Ax speed of light passes the plate without any polarization changes Putting a polarizer behind the crystal transmitting the incident polarization all frequencies given by equation 2 7 can pass through freely The free spectral 20
99. to optical trap for sodium has been achieved This diploma thesis describes the apparatus set up so far and gives an introduction to the field of ultracold degenerate Fermi gases ii ili iv Contents 1 2 Introduction 1 1 1 Quantum Statistics Bosons and Fermions 1 1 2 Degenerate Kermions zu a ee AAA 4 1 2 1 Feshbach Resonances 4 1 2 2 Theoretical Approaches 7 1 2 25 BEC Theory a SA Se Se OOO T 1 2 2 2 Strongly Coupled Fermions 8 1 2 3 Current Research Topics A Short Summary 9 13 Why Lithium AND Sodium 2000244000 Br su au 9 1 3 1 General Aspects S ok ee dd eg Bele SORE R EEO 10 1 3 2 Our Motivation for Choosing Lithium and Sodium 11 LA Outline 22 ape S net ato Goat Reet el 11 Theory 13 2A Dye Basis ee Be bee Ul 13 2 1 1 Some Laser Basics al 14 21 2 The Ring Dye ah 15 2 1 3 Singlemode Operation of Dye Lasers 18 2 1 3 1 Optical Diode OD and Thin Quartz Plate 19 2 1 3 2 Selecting a Longitudinal Mode 20 2 1 3 3 Birefringent Filter BR 20 2 1 3 4 Thin and Thick Etalon TNE and TKE 22 2 1 3 5 Tuning Resonator Modes Galvo Brewster Plate and Tweeter GP and M2 23 2 1 3 6 Locking the Laser to the Internal Fabry Perot Cav ity and Frequency Sweeps 24 2 2 Magneto Optical
100. tting the y direction the algorithm averages over all rows that are not hit by the beam This row profile is subtracted from the x lineprofile afterwards Finally the button Save current frame allows to save the current image to a Windows bitmap file The Log fitting results to file button starts logging the fitting parameters i e waist in x and y direction x and y coordinate of the center of the fitted Gaussian selected in the subsequent dialog to a user specified text file e g for an analysis of the pointing stability of the laser 68 C 3 Some Comments on the Programming and Fitting Figure 1 Live Beam Profiler File Edit View Insert Tools Desktop Window Help Waist in horizontal direction red Ogreen w 539um Oblue Waist in vertical direction Reduce stripes averaged Frames 1 w 387um df 0 vl Activate Fitting Log Fitting Results to file Save current frame Gauss Fit in horizontal direction Gauss Fit in vertical direction A mi a w bal 8 a amp 8 R 9 n intensity arb units intensity arb units a aA S A L L n 1 1 1 1500 2000 2500 3000 3500 D 500 1000 1500 2000 2500 3000 horizontal position um vertical position um g 4 5 by Stefan Weis Figure C 1 Screenshot of the Beam Profiler C 3 Some Comments on the Programming and Fit ting Since there is no Windows XP low l
101. ur combinations are reasonable and used Li SLi e g in the groups of Hulet Rice and Salomon ENS Rb K e g in the groups of Bloch Mainz and of Inguscio Florence and Li Na group of Ketterle MTT and Li 57Rb e g in the group of Zimmermann T bingen 1 3 2 Our Motivation for Choosing Lithium and Sodium We chose Li mainly for its broad Feshbach resonance that can easily be resolved So the creation of a sufficiently homogeneous magnetic field all over the entire atomic cloud is feasible As a last point Li is readily available 23Na was chosen because it is only slightly heavier as discussed before and the biggest BECs ever have been realized with sodium 48 Our Zeeman slower can be used for both species with sufficient efficiency see 3 3 1 Last but not least there are successful experiments running with Na and Li 49 So we will not have to deal with problems that have not been solved before as the target of the experiment is to reach degeneracy as soon as possible 1 4 Outline On our way towards a degenerate gas of ultracold Li atoms and a BEC of Na mainly a vacuum system and a laser system need to be set up Atoms are evap orated in two high temperature ovens slowed down trapped and cooled using magnetic fields and lasers For sodium there are no solid state lasers available at present and dye lasers need to be used The outline of this thesis is as follows 15The natural abundanc
102. ut by losses within the cavity In case different longitudinal modes are amplified by different reservoirs of excited electrons i e they do not compete the gain media are called inhomogeneously broadened for example Doppler broadening in gas lasers 2 1 2 The Ring Dye Laser In general dye molecules consist of a large number of atoms This leads to a big number of different vibrational degrees of freedom 50 atoms give rise to about 150 vibrational modes Many of these vibronic excitations directly couple to the electronic transitions adding sublevels to these spectra with typical mode spacings of approximately 1 THz to 100 THz Additionally rotational degrees of freedom come into play with a mode spacing of typically 10 GHz to 1 THz These levels are strongly broadened due to collisions with the solvent yielding a quasi continuous absorption spectrum Furthermore the band structure depends on temperature dye concentration and acid base equilibria with the molecules of the solvent There are mainly three generic classes of dye molecules cf figure 2 2 However common to all of them is the presence of several conjugated double bonds a so called system of r electrons In a simple approach one can assume the elec trons to be in a constant box potential within this system of conjugated double bonds One distinguishes linear systems e g pinacyanol circular systems e g Cu Phtalocyanin and more complicated branched systems like in th
103. ut one and a half meters When frequency sweeps are done this length needs to be adapted since otherwise mode jumps between different cavity modes would appear This feature is provided by the galvo plate also called Brewster plate a window brought into the resonator that can be turned by means of a galvanometer Turning the GP leads to the desired changes of the resonator length as the optical path inside the plate with refractive index of approx 1 5 changes The tuning range of the GP exceeds 30 GHz however the mechanical inertia inhibits fast changes and especially cannot compensate any fast fluctuations of the resonator length This is where the tweeter mirror M2 mounted onto a piezo comes into play It permits to change the resonator length quickly on the order of kHz but only with a small amplitude that corresponds to some hundreds of MHz ture like in 2 6 See any optics textbook for more details 23 Chapter 2 Theory 2 1 3 6 Locking the Laser to the Internal Fabry Perot Cavity and Fre quency Sweeps Single mode operation at maximal output power is achieved if all wavelength se lective elements mentioned before are aligned such that their transmission maxima overlap in order to minimize losses for the desired wavelength and to get a maximal frequency selectivity i e losses are substantially lower for exactly one longitudinal cavity mode than for all the others Voltage V 2 2 a 0 1 2 3 Frequenc
104. y GHz Figure 2 8 Transmission curve of the reference cavity registered by the photodiode shifted to negative voltages such that locking can easily be done on any zero crossing of the signal How this overlap is realized in practice will be described in the following First the birefringent filter is manually tuned to the approximate mode One can clearly see mode hops of 200 GHz FSR of the thin etalon on a wavemeter while turning the micrometer screw Afterwards the thin etalon offset allows to select the right frequency to about 10GHz corresponding to the FSR of the thick etalon The final position of the thin etalon is adjusted with a controller permitting to tune the center frequency of the thin etalon to 15 15 GHz The center wavelength to end up with is already now defined quite precisely The remaining elements TKE GP tweeter are synchronized electronically Therefore about 2x 1 of the outcoupled beam are split off One is directed onto a photodiode directly power signal the other one passes through a temperature stabilized and tunable Fabry Perot cavity with FSR of about 1GHz and a finesse of 2 and is captured by another photodiode This signal reference signal is divided by the power signal compensating any intensity fluctuations The thick etalon locks one peak of the etalon transmission curve to the laser wavelength This is realized using a lock in technique The thick etalon is mod ulated with a low amplitude a
105. y point refer to Jan Krieger s diploma thesis 73 3 3 1 Design Criteria e Our slower was designed for sodium As discussed before more sodium than lithium is needed for further cooling steps A sodium slower always works for lithium but is more conservative e The slower should be considerably shorter than 1m first for keeping the apparatus compact but also since the geometric divergence of the atomic beam cancels the slowing effect at some point Random scattering of the absorbed photons leads to additional transversal heating i e divergence e In order to compensate for imperfections of the magnetic field the design used a saturation parameter S equal to one even though more laser power is available e The magnetic field should not be excessively high close to the center of the MOT coils since it has to vanish there In addition the Zeeman slower beam should not be resonant to the atoms in the MOT This condition requires a non zero field at the end of the slower 3 3 2 The Setup and a Basic Introduction Our Zeeman slower consists of two solenoids with an overall length of about 70cm with an additional free space of about 20cm between them The solenoids are concentric to the atomic beam Along this axis a laser beam and the atomic beam counterpropagate Whenever the circularly polarized laser beam is resonant with an electronic transition light is absorbed and scattered into any direction leading to a mean deceleration

Setup of a Laser System for Ultracold Sodium - Towards a

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