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Dynamics of Ultracold Quantum Gases and Interferometry with Coherent Matter Waves

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Dynamics of Ultracold Quantum Gases and Interferometry with Coherent Matter Waves DYNAMICS OF ULTRACOLD QUANTUM GASES AND INTERFEROMETRY WITH COHERENT MATTER WAVES Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Naturwissenschaften der Universität Ulm vorgelegt von GERRIT PETER NANDI aus Bonn Institut für Quantenphysik Universität Ulm Leiter: Prof. Dr. Wolfgang P. Schleich Ulm 2007 Amtierender Dekan: Prof. Dr. Klaus-Dieter Spindler Erstgutachter: PD Dr. Reinhold Walser Zweitgutachter: Prof. Dr. Wolfgang Wonneberger Tag der Promotion: 25. Juni 2007 ÓÒØÒØ× ½º ÁÒØÖÓ ½ ¾º ÊÚÛ Ó Ò Ó×¹Ò×ØÒ 2.1. Quantumfieldtheoryinthemean-fieldlimit . ... 7 2.1.1. Quantizedatomicfields . 7 2.1.2. Mean-fieldapproximation . 10 2.1.3. TheGROSS-PITAEVSKII equationinscaledvariables . 13 2.1.4. Multi-component, vector-valued coupled BOSE-EINSTEIN con- densates ............................. 13 2.2. Healing length and THOMAS-FERMI limit................ 16 2.3. Low-dimensionalcondensates . 17 2.3.1. Quasione-dimensionalBEC . 18 2.3.2. Quasitwo-dimensionalBEC . 18 2.4. Hydrodynamicdescription . 18 2.5. Topological excitations – vortices in BOSE-EINSTEIN condensates . 19 2.6. Linear response analysis for multi-component BECs . ....... 20 2.6.1. One-componentBEC. 20 2.6.2. Two-componentBEC . 22 2.7. Beyond the mean-field limit – HAMILTONians with few quantized modes 23 2.7.1. TheJOSEPHSON HAMILTONian................. 23 2.7.2. Semiclassicalapproximation . 24 ¿º ÓÐ ÕÙÒØÙÑ ×× Ò Ý ¾ 3.1. Classical physics of a particle falling within the drop capsule . 28 3.1.1. Thedropcapsule ......................... 28 3.1.2. A single classical particle trapped within the drop capsule . 31 3.2. Quantum physics of identical atoms falling within the dropcapsule . 36 3.2.1. Static representation of states in EUCLIDean frames of reference 36 3.2.2. Dynamicsincomovingframesofreference . 38 3.2.3. EHRENFEST’stheorem...................... 41 3.2.4. Application of the frame transformation to mean-fieldtheory. 42 3.3. Many identical atoms within the drop capsule in a rotatingframe. 47 i Contents 3.3.1. Classical physics of the drop capsule in a rotating frame of the Earth ............................... 47 3.3.2. Asingleclassicalparticleinthedropcapsule . ..... 51 3.3.3. Manyidenticalatomsinthedropcapsule . 52 3.4. Two-componentatomicgas . 55 3.5. Conclusions................................ 56 º ÖÚØØÓÒÐ Ò ÖÓØØÓÒÐ ×Ò×Ò ÓÒ Ø ×× Ó ÑØØÖ ÛÚ Ò¹ ØÖÖÓÑØÖÝ 4.1. The atom interferometer experiment of KASEVICH and CHU ...... 60 4.2. Calculation of gravitational phase shifts in a nonlinear atom interferom- eter .................................... 63 4.2.1. Theoreticaldescriptionofthesetup . .. 63 4.2.2. The limit of a homogeneous BEC – hydrodynamic description . 65 4.2.3. Semi-classical calculation of the gravitational phaseshifts . 67 4.2.4. Discussion with respect to drop tower experiments . ..... 70 4.2.5. Inhomogeneous BEC interferometer – numerical results .... 71 4.3. Rotational sensing with quantized vortices in a BEC . ....... 73 4.3.1. Using superpositions of states with different angular momentum as a rotational sensor – basic idea on a single-particle level... 73 4.3.2. UsingtheSTIRAPschemeasabeam-splitter . 76 4.3.3. CreatingarotationalsensorbyusingSTIRAP . ... 86 º ÒÙÑ Ö ÐØÖ ÓÖ ÑØØÖ ÛÚ× 5.1. Mean-fieldcalculation . .. .. .. .. .. .. .. 90 5.1.1. HomogeneousGPequation. 91 5.1.2. Inhomogeneous BEC – numerical studies and discussion of pa- rameters.............................. 95 5.2. Beyond the mean-field approximation - HAMILTONian with two quan- tizedmodes ................................ 99 5.3. Optimizing and quantifying the effect of number stabilization . 104 5.3.1. Variation of the finalp ¯-pulse and criteria defining the region of stabilization............................ 104 5.3.2. Widthoftheoccupationdistribution . 108 5.3.3. Quadrature representation and WIGNER function . 110 º ÛØ ØÛ ÓÒÒØ ½½¿ 6.1. Scatteringtheory–ashortreview . 114 6.1.1. Simpletwo-particlescattering . 114 6.1.2. FESHBACH resonances–asimplemodel . 117 6.1.3. Examplefor a FESHBACH resonanceinaphysicalsystem . 121 ii Contents 6.2. Collective FESHBACH Scattering of a superfluid droplet on a two-component BEC.................................... 122 6.2.1. Bound BOGOLIUBOV excitationsand scattering solutions . 122 6.2.2. Collective FESHBACH scattering–stationaryapproach . 128 6.2.3. Collective FESHBACH Scattering – dynamical calculations . 133 6.3. Conclusionsandoutlook . 134 º ÑØÓ × ½¿ 7.1. Discretization of continuous problems in position and momentumspace 137 7.1.1. Discretizationinonespatialdimension . ... 137 7.1.2. Discretization of problems with cylindrical symmetry ..... 138 7.1.3. Performing calculations on the numerical grid . ..... 138 7.2. Numerical determination of the stationary solution of theGPequation . 139 7.2.1. Discrete FOURIER transformation(FT) . 139 7.2.2. Stationary solution of the GP equation via imaginary time pro- pagation.............................. 140 7.3. Dynamical solution – solving first order ordinary differential equations numerically ................................ 142 7.3.1. Predictor-correctormethods . 142 7.3.2. Numerical implementation of the time-dependent GP equation . 143 7.3.3. Solving difference equations (for two-mode problem) ..... 143 7.4. Numerical solution of the BOGOLIUBOV equations . 144 7.5. Numerical implementation of the WIGNER function . 145 º ½ º ÔÔ ÒÜ ½ A.1. Two-levelsystem ............................. 149 A.2. Cold quantum gases in micro-gravity – complements . ...... 151 A.2.1. TAYLOR expansionofthegravitationalpotential . 151 A.2.2. Canonical transformations in HAMILTONianmechanics . 152 A.2.3. Useful relations for the construction of the LIE group elements . 154 A.2.4. Classical trajectory of the drop capsule in a rotatingframe . 155 A.3. A number filter for matter waves – instantaneous-pulse approximation . 155 A.4. Frequently used symbols and constants, estimates . ........ 160 iii ½º ÁÒØÖÓ Physicists have been fascinated by the phenomenon of BOSE-EINSTEIN condensation (BEC1) since its theoretical prediction in 1924/25. In the seminal papers "PLANCKs Gesetz und Lichtquantenhypothese" [1] as well as "Quantentheorie des einatomigen ide- alen Gases" [2] S. N. BOSE and A. EINSTEIN, NOBEL Prize (NP) winner in 1921, dis- cussed the consequences of the statistical indistinguishability of atoms, which affects the thermodynamical properties of an ideal gas. A. EINSTEIN pointed out the possibility of a new kind of phase transition that occurs in such a system at ultralow temperatures and argued that this might lead to a novel state of matter. The rapid development of quantum theory since the late 1920s led to enormous progress in understanding the properties of superfluids and superconductors in the following decades. Superconductivity had been observed in 1911 already by H. KAMERLINGH ONNES (NP 1913) but was not under- stood at that time. This phenomenon could be explained by J. BARDEEN, L.N.COOPER and R. SCHRIEFFER in the 1950s (BCS-theory, NP 1972) [3]. Superfluidity of liquid 4He was discovered in 1937 independently by P. L. KAPITZA (NP 1978) on the one hand and J. ALLEN and A.D. MISENER on the other hand. In 1938 F. LONDON came up with the idea that superfluidity could be linked with BEC [4]. L. D. LANDAU (NP 1962) estab- lished his two-fluid model for helium based on quantum hydrodynamics in 1941 and was able to explain the spectrum of elementary excitations [3]. The first microscopic theory of interacting BOSE gases was developed by N. N. BOGOLIUBOV in 1947 [5], who pre- sented a derivation of the corresponding phonon spectrum and introduced the mean-field description. Quantized vortices in superfluids were predicted byL. ONSAGER as well as R. P. FEYNMAN (NP 1965), and were observed experimentally in the 1950s [3, 6]. With the derivation of a mean-field equation by E. P. GROSS and L. P. PITAEVSKII in 1961 [7, 8], the dynamics of the macroscopic wave function could be described in terms of a nonlinear SCHRÖDINGER-like equation. While the preceding considerations mostly concerned the condensed matter commu- nity, advanced mechanisms for cooling neutral particles became feasible since the 1970s in atomic physics. By 1985, atomic beams could be slowed down by laser cooling, which had been suggested by T. W. HÄNSCH (NP 2005) already in 1975. Thus, physicists were able to cool down atoms to temperatures below 1 mK, and optical molasses could be produced. Two years later, the first magneto-optical traps (MOT) for laser-cooled atoms were established. These developments led to the NOBEL Prize in physics for S. CHU, 1We will use the acronym "BEC" both for the terms "BOSE-EINSTEIN condensation" as well as "BOSE- EINSTEIN condensate". 1 1. Introduction C. COHEN-TANNOUDJI and W. D. PHILLIPS in 1997. Dilute atomic gases differ from systems like liquid helium by their lower densities and collisional rates. Using the cool- ing techniques described above as well as evaporative cooling, spin-polarized hydrogen seemed to be the natural candidate to condense, as it remains gaseous down to ultralow temperatures. In the 1980s, J. T. M. WALRAVEN as well as D. KLEPPNER made large progress on this area but failed to demonstrate BEC with spin-polarized hydrogen. For a historical overview of the cooling techniques, see for example [9, 10, 11]. Finally, by the means of evaporative cooling the critical temperature (µK/nK-range) and phase-space density for the onset of BOSE-EINSTEIN condensation could be reached in dilute alkali vapors and was demonstrated in the laboratory in 1995 for the first time by the groups of E.
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