Atomic Bose-Einstein condensates and Rydberg gases Sebastian Wüster Summer-term 2013 Information contact details: Sebastian Wüster, • Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38, Dresden Office: 2B8 email://[email protected] Phone: +49 351 871 2208 web: http://www.mpipks-dresden.mpg.de/~sew654/teaching.html • Lecture: Wednesdays: 16:40 - 18:10 Room BZW/ A120/ P 13 lectures (10.4. - 17.7., not 22.5, 5.6.) beamer introduction, then chalk and blackboard + movies and images Literature (1) C.J. Pethick and H. Smith, “Bose-Einstein Condensation in Dilute Gases” , Cambridge University Press (2002) (2) L.P. Pitaevskii and S. Stringari “Bose-Einstein Condensation”, Oxford University Press (2003) (3) F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari “Theory of Bose- Einstein condensation”, Rev. Mod. Phys. 71, 463 (1999) (4) T. F. Gallagher, “Rydberg atoms”, Cambridge University Press (1994) Outline 1. Introduction 1.1. Motivation 1.2. Revision 2. Ultra-cold atomic gases 2.1. Quantum statistical physics 2.2. Trapping and cooling of atoms 2.3. Interactions between atoms 3. Bose-Einstein condensates 3.1. Field operator 3.2. Gross-Pitaevskii theory 3.3. Excitations of the condensate 4. Rydberg atoms 4.1. Quantum defects 4.2. Rydberg interactions and external fields 4.3. Rydberg excitation and decay 4.4. Dipole blockade 4.5. Field ionization 4.6. Applications in quantum information 1. Introduction • Purpose of 1.1. motivation: ★ get you excited about the field of ultra-cold atoms (quantum-atom-optics) ★ show for what the material presented later is important ★ personal view: highlight use of BEC for inter-disciplinary physics (quantum simulation) • Purpose of 1.2. revision: ★ remind you of required material from quantum mechanics and atomic physics ★ make sure everyone has similar starting point ★ will not replace prior study of these topics, if you notice that you are not familiar with some of the physics, please recap. Survey: Please indicate for the following physics topics how familiar you feel that you are with them: never not had in remember what it is, heard and can fully heard lecture but no details revise the details familiar quantum oscillator: second quantisation: quantum field theory: quantum statistical physics: Bose-Einstein statistics: Zeeman effect: Stark effect: two-level atom interacting with light field: hyperfine splitting: quantum scattering-theory/ partial wave expansion: 1.1. Motivation (part1) Bose-Einstein Condensation Bose 1924: Identical quantum particles (e.g. photons) have to be treated as indistinguishable. This affects the counting of the possible ways to realize a given state. S. N. Bose, Z. Phys. 26 (1924) 178. Albert Einstein Satyendra Nath Bose Einstein 1925: This results at T=0 in a case where all bosons occupy the same quantum state. A.Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. page 261 (1924). A.Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. page 3 (1925). 1 => Bose-Einstein Statistics: Occupation of state with energy E: n = e(E µ)/kB T 1 − − => Bose-Einstein Condensation: At T=0, all Bosons will occupy the same lowest energy quantum state cloud was determinedfrom the absorption tion to characterizing any deviations from fractions, respectively. (Figs. 2B and 4). As of a 20-is, circularlypolarized laser pulse thermal equilibrium. The measurement the cooling progresses (Fig. 4), the noncon- resonantwith the 5S1/2,F - 2 to 5P3/2, F - process destroys the sample, but the entire densate fraction is reduced until, at a value 3 transition.The shadowof the cloud was load-evaporate-probe cycle can be repeat- of vevap of 4.1 MHz,little remainsbut a pure imagedonto a charge-coupleddevice array, ed. Our data represent a sequence of evap- condensate containing 2000 atoms. digitized,and storedfor analysis. orative cycles performed under identical The condensate first appears at an rf This shadow image (Fig. 2) contains a conditions except for decreasing values of frequency between 4.25 and 4.23 MHz. The large amount of easily interpreted infor- vevap, which gives a corresponding de- 4.25 MHz cloud is a sample of 2 X 104 mation. Basically, we did a 2D time-of- crease in the sample temperature and an atoms at a number density of 2.6 x 1012 flight measurementof the velocity distri- increase in phase-space density. cm-3 and a temperature of 170 nK. This bution. At each point in the image, the The discontinuous behavior of thermo- represents a phase-space density pps of 0.3, optical densitywe observedis proportional dynamic quantities or their derivatives is which is well below the expected value of to the column density of atoms at the always a strong indication of a phase tran- 2.612. The phase-space density scales as the correspondingpart of the expandedcloud. sition. In Fig. 3, we see a sharp increase in sixth power of the linear size of the cloud. Thus, the recorded image is the initial the peak density at a value of vevap of 4.23 Thus, modest errors in our size calibration velocity distribution projected onto the MHz. This increase is expected at the BEC could explain much of this difference. Be- plane of the image. For all harmoniccon- transition. As cooling proceeds below the low the transition, one can estimate an fining potentials, includingthe TOP trap, transition temperature, atoms rapidly accu- effective phase-space density by simply di- the spatial distributionis identical to the mulate in the lowest energy state of the 3D viding the number of atoms by the observed velocity distribution,if each axis is linear- harmonic trapping potential (23). For an volume they occupy in coordinate and ve- ly scaled by the harmonic oscillator fre- ideal gas, this state would be as near to a locity space. The result is several hundred, quency for that dimension (22). Thus, singularity in velocity and coordinate space which is much greater than 2.6 and is con- from the single image we obtained both as the uncertainty principle permits. sistent with a large occupation number of a the velocity and coordinate-spacedistri- Thus, below the transition we expect a single state. The temperatures and densities butions, and from these we extracted the two-component cloud, with a dense central quoted here were calculated for the sample temperatureand central density, in addi- condensate surrounded by a diffuse, non- in the unexpanded trap. However, after the condensate fraction.Experimental This behavior is clear- adiabatic expansion Condensation stage, the atoms are in dilute ly displayed in sections taken horizontally still in good thermal equilibrium, but the 4I I II through the center of the distributions, as temperatures and densities are greatly re- 4~~~~~~~~~~~~~~~~~~~~~~~~~~shown in Fig. 4. For values of vevap above duced. The 170 nK temperature is reduced 4.23 MHz, the sections show a single, to 20 nK, and alkalithe number density gases is reduced smooth, Gaussian-like distribution. At 4.23 from 2.6 X 1012 cm3 to 1 x 1011 cm-. theoreticalMHz, a sharp central prediction peak in the of distribu- BEC 1925There is no obstacle to adiabatically coolingNobel Prize in Physics (2001), E 3 tion begins to appear. At frequencies below and expanding the cloud further when it is en) 1111 experimental4.23 MHz, two distinct realization components to 1995 the desirable to reduce the atom-atom interac-W. Ketterle ,E.A. Cornell and C.E. Wieman E 0 cloud are visible, the smooth broad curve tions, as discussed below (24). and a narrow central peak, which we iden- A striking feature evident in the images ' 2 tify as the noncondensate and condensate shown in Fig. 2 is the differing axial-to- radial aspect ratios for the two components 7,_I of the cloud. In the clouds with no conden- 4.71 sate (vevap > 4.23 MHz) and in the non- condensate fraction of the colder clouds, the velocity distribution is isotropic (as ev- 01 idenced by the circular shape of the yellow 4.25 to green contour lines in Fig. 2, A and B). But the condensate fraction clearly has a 4.23 4.0 4.2 4.4 4.6 4.8 larger velocity spread in the axial direction than in the radial direction (Fig. 2, B and Vevap(MHZ) 4.21 C). This difference in aspect ratios is readily Fig. 3. Peak density at the center of the sample as 4.19 explained and in fact is strong evidence in a function of the finaldepth of the evaporative cut, support of the interpretation that the cen- vevap. As evaporation progresses to smaller values tral peak is a Bose-Einstein condensate. The of vevap' the cloud shrinks and cools, causing a f 4.16 modest increase in peak density untilvevap reach- noncondensate atoms represent a thermal es 4.23 MHz. The discontinuity at 4.23 MHz indi- distribution across many quantum wave cates the first appearance of the high-density functions. In thermal equilibrium, velocity condensate fraction as the cloud undergoes a 4.11 distributions of a gas are always isotropic phase transition.When a value for Vevap of 4.1 Mhz regardless of the shape of the confining is reached, nearly all the remaining atoms are in potential. The condensate atoms, however, the condensate fraction. Below 4.1 MHz,the cen- are all describedK.B. by theDavis same wave et function,al., Phys. Rev. Lett. 75 (1995) 3969. traldensity decreases, as the evaporative "rfscal- which will have an anisotropy reflecting 4.06 pel" begins to cut into the condensate itself. Each that of the confining potential. The veloc- data point is the average of several evaporative M. H. Anderson et al. , Science 269 (1995) 198. 300 pm of the wave function cycles, and the error bars shown reflect only the ity spread ground-state scatter in the data.