a general quantum-engineering technique for efficient

A general quantum-engineering technique for efficient production of ultracold dipolar moleculesLukas K. Reichsöllner
October 11, 2017
PhD thesis
by
October 11, 2017
meiner Familie
v
Abstract
This thesis reports on an universal method for highly efficient production of a low entropy ensemble of heteronuclear molecules. The goal is to efficiently "quantum-engineer" a strongly dipolar quantum gas in an optical lattice with unity filling. First, the simultaneous production of two quantum degenerate samples of 87Rb and 133Cs is revisited in view of the subsequent manipulation steps. Then we load the two spatially separated Bose- Einstein condensates of Rb and Cs into an optical lattice and we create a Mott insulator of Cs with exactly one atom per lattice site to suppress loss. The lattice depth is adjusted to simultaneously support a superfluid of Rb which is of fundamental importance because in the next step we move Rb towards the Cs Mott insulator with the movable Rb trap. A magnetic Feshbach resonance is used to tune the interspecies interactions to zero and to merge both atomic samples. In combination with the repulsive Rb-Rb interactions, Rb-Cs precursor pairs can form on each individual lattice site. An adiabatic magnetic-field ramp associates the paired atoms to RbCs molecules. Coherent adiabatic transfer by stimulated Raman adiabatic passage (STIRAP) can be applied to subsequently transfer the weakly-bound molecules to the rovibronic ground state. An electrical field can be used to polarize the ensemble and to tune the long-range dipole-dipole interactions This work succeeded in producing an outstanding filling fraction of more than 30% of molecules in an optical lattice. This value can be pushed much higher by several improvements, which will be discussed in this work.The method developed in this thesis provides a system with tunable long-range interactions, perfectly suited for quantum- simulation experiments, tests of fundamental constants and investigations of novel states of matter.
vii
Zusammenfassung
In dieser Arbeit berichte ich über eine universell einsetzbare Methode zur Erzeugung eines Ensembles mit niedriger Entropie aus heteronuklearen Molekülen. Die Zielsetzung ist die quantenphysikalische Konstruktion eines ausgeprägt dipolaren Quantengases mit einem einzel- nen Teilchen an jedem Gitterplatz. Zuerst steht die überarbeitete, zeitgleiche Produktion zweier quantenentarteter Ensembles von 87Rb und 133Cs unter Berücksichtigung der folgenden Manipulationsschritte im Vordergrund. Danach laden wir die beiden räumlich getrennten Bose-Einstein Kondensate aus Rb und Cs kontrolliert in ein optisches Gitter, sodass wir einen Cs Mott Isolator mit genau einem Cs Atom pro Gitterplatz erschaffen um Teilchenverluste zu verhindern. Die Gittertiefe wird so hochgefahren, dass sich zeitglich ein Rb Superfluid im Gitter ausbilden kann, was fundamental für den nächsten Schritt ist, da in diesem das superfluide Rb mit einer beweglichen Dipolfalle in Richtung dem Cs Mott Isolator bewegt wird und schließlich mit diesem zur vollständigen Überlagerung gebracht wird. Dazu wird auch eine magnetische Feshbach Resonanz eingesetzt um die interspezies Wechselwirkung auszuschalten und die Überlagerung überhaupt zu ermöglichen. Durch die außerdem gegebene repulsive intraspezies Wechselwirkung bei Rb is es möglich genau ein Rb-Cs Atompaar pro Gitterplatz zu erzeugen, sodass eine adiabatische Magnetfeldrampe angewendet werden kann um aus den Rb-Cs Paaren RbCs Moleküle zu erzeugen. Anschließender kohärenter adiabatischer Transfer durch stimulated Raman adiabatic passage (STIRAP) ermöglicht den Transfer der schwach gebun- denen Moleküle in den rovibronischen Grundzustand. Mittels eines elektrischen Feldes kann das Ensemble polarisiert und die Stärke der langreichweitigen Dipol-Dipol Wechselwirkung eingestellt werden.
Diese Arbeit konnte erfolgreich ein bisher unerreichtes Gitterfüllungsverhältnis mit Molekülen von über 30% nachweisen. Es wird erwartet, dass dieser Wert durch einige Verbesserungen, die ebenfalls in dieser Arbeit beschrieben werden, weiter gesteigert werden kann. Die hier vorgestellte Methode ermöglicht die effiziente Erzeugnung eines Systems mit einstellbarer, langreichweitiger Wechselwirkung, welches sich perfekt für Quantensimulationsexperimente, Tests von Naturkonstanten und Untersuchung neuartiger Materiezuständen eignet.
ix
Contents
1. Quantum engineering of novel systems 1 1.1. Fundamental research with ultracold systems . . . . . . . . . . . . . . . . . . 1
1.1.1. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2. Simulating matter with ultracold gases . . . . . . . . . . . . . . . . . . 1
1.2. Supersolidity and the extended Bose Hubbard model . . . . . . . . . . . . . . 3 1.3. Dipolar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4. Creation of ultracold dipolar molecular ensembles . . . . . . . . . . . . . . . . 7 1.5. Thesis goals and thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Overview of cooling Rb and Cs 11 2.1. Oven and Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2. Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3. Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4. Dual species Raman sideband cooling (DRSC) . . . . . . . . . . . . . . . . . . 14 2.5. Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1. Optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5.2. Evaporation and Bose-Einstein condensation . . . . . . . . . . . . . . . 16 2.5.3. The s-wave scattering length as . . . . . . . . . . . . . . . . . . . . . . 17
3. Improvements of the experimental setup 19 3.1. Optical dipole trap setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1. Old dimple trap setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2. New dimple trap setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2. Magnetic-field setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1. Main magnetic-field coils . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2. Tracking-coil system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.3. Magnetic hum-cancellation system . . . . . . . . . . . . . . . . . . . . 25
3.3. Adjusted simultaneous evaporation of Rb and Cs . . . . . . . . . . . . . . . . 27 3.3.1. Bose-Einstein condensate . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.2. Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.3. Evaporation benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4. Characterization of the Rb and Cs BECs . . . . . . . . . . . . . . . . . 32
4. An optical lattice for RbCs-molecule production 37 4.1. Technical setup of the optical lattice . . . . . . . . . . . . . . . . . . . . . . . 37 4.2. Overview of the band structure of the simple cubic lattice . . . . . . . . . . . 38 4.3. Wannier states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4. Superfluid and Mott insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.1. Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4.2. Ground states of the Bose-Hubbard model . . . . . . . . . . . . . . . . 43 4.4.3. Superfluid to Mott insulator quantum-phase transition . . . . . . . . . 44
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4.5. Loading of an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5.1. Non-adiabatic loading of an optical lattice . . . . . . . . . . . . . . . . 46 4.5.2. Adiabatic loading of an optical lattice . . . . . . . . . . . . . . . . . . . 47
5. Lattice characterization 49 5.1. Lattice-depth calibration with short lattice pulses . . . . . . . . . . . . . . . . 49 5.2. Adiabatic loading of the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3. Depth calibration with lattice depth modulation technique . . . . . . . . . . . 53 5.4. Characterization of the Cs MI with Cs2 molecules . . . . . . . . . . . . . . . . 57 5.5. Characterization of the Cs MI with lattice-depth modulation spectroscopy . . . 59 5.6. Characterization of the Cs MI with the Lattice-Tilt Method . . . . . . . . . . . 62 5.7. Lifetime of the Cs MI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6. Merging of the Rb and Cs samples 69 6.1. Cs MI lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2. Maximum transport speed in an optical lattice . . . . . . . . . . . . . . . . . . 69 6.3. Transport and adiabatic merging of a double well . . . . . . . . . . . . . . . . 70 6.4. Possible issues of SF Rb transport . . . . . . . . . . . . . . . . . . . . . . . . . 73
7. Rb-Cs Feshbach resonances and RbCs-molecule production 75 7.1. Magnetic Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2. Accessible MFRs and molecule-production paths . . . . . . . . . . . . . . . . . 76 7.3. Investigations of the pole and width of a MFR . . . . . . . . . . . . . . . . . . 79
7.3.1. Loss spectroscopy in an optical lattice . . . . . . . . . . . . . . . . . . 80 7.3.2. Dependence of cloud position and size on as . . . . . . . . . . . . . . . 85 7.3.3. Conservation of Rb superfluidity . . . . . . . . . . . . . . . . . . . . . 87
7.4. Choice of the best MFR for molecule production . . . . . . . . . . . . . . . . . 89 7.4.1. Magneto-association of Rb-Cs atom pairs . . . . . . . . . . . . . . . . . 90 7.4.2. Molecule-production sequence . . . . . . . . . . . . . . . . . . . . . . 92
7.5. Feshbach molecules in an optical lattice . . . . . . . . . . . . . . . . . . . . . 94 7.5.1. Characterization of molecule-production efficiency . . . . . . . . . . . 94 7.5.2. Evaluation of lattice filling and entropy . . . . . . . . . . . . . . . . . . 98
8. Towards unity lattice filling with ground-state molecules 101 8.1. General method for efficient creation of ground-state molecules . . . . . . . . 101 8.2. Improvements for the production efficiency of RbCs molecules . . . . . . . . . 103
8.2.1. Magnetic field improvement . . . . . . . . . . . . . . . . . . . . . . . . 104 8.2.2. 3D lattice conveyor belt . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.2.3. Blue-detuned barrier dimple . . . . . . . . . . . . . . . . . . . . . . . . 106
8.3. Ground-state transfer, STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A. Appendix 113 A.1. Design of the magnetic-field coil driver . . . . . . . . . . . . . . . . . . . . . . 113 A.2. Calibration of the Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.3. Blow-away technique, high-field absorption imaging . . . . . . . . . . . . . . 121 A.4. Labeling of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.5. Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography 125
Acknowledgements 139
1Quantum engineering of novel systems
In this thesis I report on the realization of a general and highly efficient scheme for the creation of quantum gases that consist of ultracold ground-state molecules. Dipolar quantum gases provide great opportunities for the investigation of a broad range of phenomena in condensed matter and many-body physics. Our scheme produces dense and low entropy samples of RbCs molecules, starting from two quantum degenerate atomic ensembles of Rb and Cs. This thesis reports on the application of quantum mechanics to "quantum-engineer" a molecular many-body system.
1.1 Fundamental research with ultracold systems 1.1.1 Historical remarks
The research with cold (sub-Kelvin) and ultracold (sub-Millikelvin) matter was outstandingly successful in the past few decades. Several Nobel prizes were awarded for the closely related research fields of superfluidity, conductivity and superconductivity. In 1987 J. G. Bednorz and K. A. Müller received the Nobel prize for the "...discovery of superconductivity in ceramic metals". In 1998, R. B. Laughlin, H. L. Störmer and D. C. Tsui got the Nobel prize for the "discovery of a new form of quantum fluid with fractionally charged excitations" and in 1996 D. M. Lee, D. D. Osheroff and R. C. Richardson shared the Nobel prize for the "...discovery of superfluidity in helium-3". Again, in 2003 the Nobel committee decided on giving the Nobel prize for the "...pioneering contributions to the theory of superconductors and superfluids" to A. A. Abrikosov, V. L. Ginzburg and A. J. Leggett. All of these works have in common that the discovered phenomena take place in complex systems and are about interactions and transport of particles (conductivity) at very low temperatures. R.P.Feynman introduced the idea of simulating such complex quantum systems by using controllable and known quantum systems [Fey82].
One possible realization of a "quantum simulator" are ultracold gases. Research with ultracold and quantum degenerate samples became possible in the last two decades thanks to the "development of methods to cool and trap atoms with laser light", and for this work the Nobel prize was awarded to St. Chu, C. Cohen-Tannoudji and W. D. Phillips in 1997. Laser cooling allowed work with ultracold atoms and for "the achievement of Bose-Einstein condensation in dilute gases of alkali atoms..." the Nobel prize was awarded to E. A. Cornell, W. Ketterle and C. E. Wieman in 2001. The realization of Bose-Einstein condensates (BEC) and quantum degenerate fermionic systems was crucial for making quantum simulation of solid state systems possible. "Quantum engineering", rephrased by M. Lewenstein "preparation, manipulation, control and detection of quantum systems" [Lew07], provides a bottom-up approach to tailor such ultracold quantum gases for quantum simulation. Quantum engineering with ultracold atoms can even exceed the possibility of solid state systems due to the arbitrary choice of bosonic or fermionic constituents, tunable interactions, adjustable periodic potential, variable external confinement and full control of all internal and external degrees of freedom.
Ultracold quantum systems can be used not only for quantum simulation [Mos16; Ort09] but also for tests of the standard model of particle physics [Hun12], investigations of fundamental constants [Chi09], research of novel states of matter [Gre02; Jor08; Sch08; Gór02; Cap10; Yi07], quantum information processing [Ort11] and quantum chemistry, i.e chemistry where quantum statistics, quantum correlations and partial waves govern chemical processes.
1.1.2 Simulating matter with ultracold gases A huge variety of applications of ultracold systems exist (cf. e.g. Ref. [Lew07]). One
example is the superconductivity in metals ("conventional superconductivity"). Conventional
1
superconductivity is explained by the Bardeen-Cooper-Schrieffer (BCS) theory [Bar57]. The fermionic pairing in the BEC-BCS crossover regime and the entering of a superfluid state was studied with trapped ultracold fermionic atoms [Reg04; Bar04]. Superfluidity is a state of matter when flow occurs without loss of kinetic energy. However, a full explanation for unconventional superconductivity and in particular high-temperature superconductivity is still missing and several theories (e.g. in Ref. [Zha97]) and research routes are debated [Zaa06]. From P.W. Anderson’s contribution in Ref. [Cla03] it appears established that the understanding of the 2D Hubbard model [Aue12] holds at least some explanations of high-temperature superconductors. The Hubbard model explains the transition from insulating to non-insulating states of interacting particles in a lattice. Almost perfect realizations of many different Hubbard models [Jak05] can be made with ultracold atomic gases confined in lattices.
The Bose-Hubbard Model (BHM), named after the type of involved particles (Bosons) and the British physicist John Hubbard, describes the properties of particles which are confined to the lowest state of a periodic structure (cf. Chap. 4.4.1). A detailed description of the BHM can be found in Refs. [Jak98; Jak05; Mor06]. In this model, particle interactions are restricted to contact-type interactions. The BHM assumes that all relevant energies are much smaller than the band-gap energy, i.e. the energy difference to the first excited Bloch band [Jak98]. The situation where only the lowest lattice band is populated is also called the tight-binding approximation.
The bosonic and fermionic Hubbard model was realized in ultracold atomic systems by using "crystals of light". Such a crystal is a perfectly periodic structure, generated by two or more counterpropagating laser beams. The anti-nodes of the created standing wave are populated with atoms. A few prominent research examples with crystals of light are the quantum-phase transition from a SF to a Mott Insulator (MI) [Gre02], observation of Bloch oscillations [Ben96] and super-Bloch oscillations [Hal10], super-exchange interactions [Tro08], repulsively-bound atom pairs [Win06], etc. All of the mentioned examples rely on the same premises: quantum degenerate samples that are confined in different well-defined and periodic trapping geometries with the possibility of tuning the atomic interactions. In systems with alkali atoms the particle interaction is restricted to contact-type interactions, represented by a delta-like pseudopotential Vint = gδ(r − r′) where g = 4π~2as/m denotes the coupling parameter with the scattering length as and particle mass m. All other interactions are negligible. The presence of a magnetic (Fano-)Feshbach resonances (MFR) [Mie00; Chi10] allows to control the short-range interaction by varying the s-wave scattering length as. A MFR uses the avoided crossing of an atomic and a molecular state and does not only allow tuning of as but also production of weakly bound molecules, hence the name Feshbach molecules. The extension of the standard BHM to long-range interactions [Bar08] and its realization is a main research focus of ultracold quantum gases.
d r
d
θ
z
Figure 1.1.: Sketch of two dipoles with dipole moment d aligned along the vertical (z) axis. The dipoles are separated by a distance r and the dipole orientation with respect to the line of sight from one dipole to the other inclines an angle Θ. The dipole moment can be either a magnetic dipole moment µ or an electric dipole moment del.
Tunable long-range interactions can be introduced by the dipole-dipole interaction. In current ultracold model systems tunable and strong long-range interactions are absent. The
2 Chapter 1 Quantum engineering of novel systems
quest for identifying a suitable interaction type with long-range character and a feasible implementation scheme has put forward the dipole-dipole interaction. The dipole-dipole interaction appears perfectly suited for this task due to its 1/r3 power law. Controlling the azimuthal angle Θ (Fig. 1.1), that is inclined by the line-of-sight between two dipoles r and the orientation of the electric dipole moment del, allows tuning of the interaction strength via an external electric field. The interaction of a magnetic dipole with magnetic dipole moment µ can be tuned by an external magnetic field. The interaction potential of polarized dipoles along the z-axis reads
Vdd(r) = Cdd
r3 . (1.1)
The parameter Cdd represents either the electric dipolar coupling Cdd = d2 el/ε0 or the magnetic
dipole-dipole coupling Cdd = µ0µ 2, where ε0 is the vacuum permittivity and µ0 is the vacuum
permeability. Comparing the dipole interaction potential of magnetic and electric dipoles, one finds that electric dipoles are generally a factor µ0µ
2/(d2 el/ε0) ∝ α2 ≈ 1 · 104 stronger, where
α = 1/137 is the fine structure constant. The total interaction potential Vint(r) is given by the sum of the short-range potential and the dipole-potential
Vint(r) = 4π~2a
m δ(r) + Cdd
r3 , (1.2)
where a represents the scattering length as explained in Ref. [Bar12]
1.2 Supersolidity and the extended Bose Hubbard model In ultracold dipolar systems, many ingredients of condensed matter systems are present: a
periodic potential and particles that are mobile or immobile, depending on the interplay of the interactions and the confinement. Additionally, ultracold quantum gases allow full control over all external and internal degrees of freedom. Arbitrary adjustment of the interplay of all relevant energy scales, interactions and correlations is possible in specific ultracold dipolar systems
New intriguing properties of ultracold dipolar systems are predicted like the coexistence of superfluidity and spatially periodic modulation of the density. This gives rise to a new type of matter: supersolids. The supersolid phase can be understood with the extended Bose Hubbard model (eBHM). It is based on the BHM. A more detailed discussion of the BHM can be found in Chap. 4.4.1 and references therein. Both models, BHM and eBHM, describe a many-particle system with a periodic potential but the type of particle interaction differs.
Here, I will briefly review the BHM, where particles are confined in a periodic structure and the interactions are constrained to on-site interactions. The Hamiltonian used to describe such ultracold systems confined to an optical lattice is composed of three terms [Jak98]:
H = ∫ d3r ψ†(r)
∫ d3r ψ†(r)ψ†(r)ψ(r)ψ(r). (1.3)
The first term represents the kinetic energy, the second term the total potential, comprising an (optical) lattice potential and an external confinement, and the last term describes the contact interactions. The particle mass is m. By using the Wannier function of the lowest lattice band w(r − rj), annihilation and creation operators bj bj , and the number operator ηj the Hamiltonian can be rephrased (cf. Chap. 4.4.1):
HBH = −J ∑ j,j′
ηj(ηj − 1) + ∑ j
εj ηj . (1.4)
The Hamiltonian HBH describes the situation where a particle can go from a lattice site j to a
1.2 Supersolidity and the extended Bose Hubbard model 3
neighboring site j′ = j + 1 with the tunnel-coupling parameter J . A particle experiences the on-site interaction U on lattice site j with ηj − 1 other particles. The external confinement gives rise to an energy offset εj on lattice site j. This notation of the standard BHM is useful for understanding the formation of an insulating state, a MI, with integer particle number ηj on a lattice site. This state is induced when the on-site interaction dominates over the tunneling and the particles are localized at the lattice sites. If U is very small compared to J then a superfluid state (SF) is present.
Adding long-range interactions to this system changes the ground-state properties. For 2D systems one can use Eq. 1.2 for describing the long-range interactions [Gór02]. Assuming that the dipole-dipole interaction can be made sufficiently strong across several lattice sites, the Hamilton operator from Eq. 1.4 has to be adjusted. The interaction term that has captured so far only on-site interactions has to be replaced [Gór02]:
1 2U
∑ j
(1.5)
Here, U0 is the on-site interaction, U1 is the nearest-neighbor interaction, U2 is the next-nearest neighbor interaction, etc. By replacing the interaction term in Eq. 1.4 by Eq. 1.5 as indicated, one obtains the Hamiltonian of the eBHM. Extensive studies of lattice systems with long-range interactions are reported e.g. in Refs. [Cap10; Mic07; Büc07]. The additional interaction terms have fundamental impact on the quantum phases that emerge in these systems. The supersolid phase for example shows on a short-range scale a periodic pattern, similar to crystalline solids, but there is also dissipationless flow of particles, which gives rise to superfluid behavior.
In Fig. 1.2 the phase diagram for dipolar particles confined to a 2D lattice at T = 0 and without an external confinement is shown. The vertical axis shows the chemical potential µc in units of the repulsive long-range interaction V and horizontal axis represents the tunnel coupling J in units of V . The phase diagram in Fig. 1.2(a) shows fractional fillings of 1/2, 1/3, 1/4. The regions with fractional fillings are surrounded by the supersolid (SS) phase. The lattice filling of 1/2 is also called "checkerboard phase", shown in the illustration in Fig. 1.2(b), where the red dots represent dipolar particles confined to an individual lattice site and the white blanks are lattice vacancies. The 1/3-filling representation in Fig. 1.2(c) is called "stripe phase" and the 1/4-filling is referred to as "star phase".
0 0.1 0.2 0.3 1
2
3
4
5
6
1/2
1/3
1/4
(a)
SS
SS
SS
SS
DS
DS
SF
J/V
(b)
(c)
(d)
Figure 1.2.: Illustration of theoretical calculations for the phase diagram of dipolar particles in a 2D optical square lattice. The vertical axis shows the chemical potential µc in units of the (repulsive) long-range interaction V , the horizontal axis denotes the tunneling coupling J in units of the long-range interaction parameter. (a) A new stable state emerges for non- integer lattice filling ηj = 1/2. It is surrounded by the supersolid (SS) phase, indicated by the green area. This state is often referred to as the checkerboard phase, illustrated in (b). A red dots represent a dipolar particle in the lattice and white blanks are vacancies. Below this stable region other states emerges with different fractional fillings at ηj = 1/3 and ηj = 1/4, surrounded by another SS shell, the corresponding filling pattern is given in (c) and (d). Image taken from Ref. [Cap10]
1.3 Dipolar systems In the previous section I have pointed out the importance of long-range interactions and
I will now discuss their implementation using three examples for ultracold dipolar systems: Rydberg atoms, magnetic atoms and, last but not least, dipolar molecules. My thesis focuses on ultracold dipolar molecules which are probably the most promising dipolar system.
For comparison of the interaction strength of the electric and magnetic dipoles, the characteristic radius add is very often used. The quantity add has the unit of length and is defined as
add = Cddm
12π~2 , (1.6)
with the particle mass m. Also, the typical distance between the dipole, defined by the optical lattice, must be considered. Many quantum gas experiments, and also the Rb-Cs setup in Innsbruck use a simple cubic (SC) lattice with a lattice constant alatt = 532 nm≈ 1 · 104 a0. For comparison, 87Rb is commonly used in BEC experiments and would have a characteristic radius aRb
dd = 0.7 a0 due to its dipole moment of µ = 1 µB. Highly excited atoms, so-called Rydberg atoms, show a very strong electric dipole moment
del ∝ nat 2, with the principal quantum number nat. This leads to strong dipolar interaction
energies, scaling with nat 4. The characteristic radius add of Rydberg atoms can be several
orders of magnitude larger than add of polar molecules or magnetic atoms. However, the lifetime of those Rydberg states are comparatively short, ∝ nat
−3. The lifetime gets even more reduced when the Rydberg atom experiences the strong force of other particles [Li04]. The short lifetime, typically in the millisecond regime, is problematic because it is on the same order as dynamical processes like tunneling, so I will not discuss them in more detail here.
Another option is to use atoms with a large permanent magnetic dipole moment like Cr, Er, and Dy. The great advantage of magnetic atoms is the presence of closed optical transitions. This allows cooling by conventional laser cooling methods and evaporation to quantum
1.3 Dipolar systems 5
degeneracy can be applied [Gri05; Aik12; Lu11]. All three species have been brought to quantum degeneracy as reported in Refs. [Gri05; Aik12; Lu11], respectively. The magnetic dipole moment µ of magnetic atoms is usually between µ = 6µB (µB is the Bohr magneton) for 52Cr and µ = 10µB for 164Dy.
The characteristic radius of magnetic atoms is much shorter than the typical lattice spacing. For example, aCr
dd = 16 a0 for Cr and for the most magnetic atom, Dy, aDydd = 132 a0. Dipolar effects of magnetic atoms were already investigated in 3D traps [Lah08; Kad16]. In the first reference the d-wave dynamics of a collapsing 52Cr BEC was investigated. The second work explored the dynamics when the ratio of the long-range interactions and short-range interactions was varied. A finite-range instability was created in an ultracold gas of 164Dy by tuning the short-range interactions via the s-wave scattering length. The authors directly observed the formation of a triangular structure of droplets.
In optical lattices dipolar effects were observed with magnetic atoms [Bai16] as well. The authors created a MI of 168Er atoms. They investigated the on-site interaction in presence of magnetic dipolar long-range interactions, depending on the dipole-dipole orientation, and the SF-MI transition. The distance between the magnetic dipoles due to the lattice spacing makes the magnetic interaction comparatively weak. Thus, the small magnetic interaction poses an experimental difficulty, since it has to be on the same order of magnitude as the other energy scales which are set by the on-site interaction or the temperature. Still, atoms with a large magnetic dipole moment have proven to be a very useful tool for research with dipolar systems.
An electric dipole moment is absent in all ground-state atoms. Compounds that would provide pronounced electric dipole moments of up to d ∼ 10 D are heteronuclear dimer molecules in their absolute, i.e. rovibronic, ground state. Rovibronic is a blend word and stands for rotational vibrational and electronic. The large dipole moments of such ground- state dimers have already been measured for several candidates, e.g. dKRb = 0.57 D[Ni08], dRbCs = 1.17 D [Mol14; Tak14]. For these dimers, the characteristic range is typically between aKRb
dd = 4 · 103 a0 and aLiCs dd ≈ 4 · 105 a0, but the most relevant value here is related to the RbCs
molecule: aRbCs dd = 2.8 · 104 a0. Note, that for any of the mentioned examples here add ≥ alatt
Great success with the work with dipolar diatomic ground-state molecules has been reported so far by the JILA group [Mos15a] (cf. also overview article [Mos16]) with 40K87Rb. The work with dipolar molecules is more challenging than with magnetic atoms due to the more complex internal structure and the absence of closed cycling transitions for laser cooling. But the additional degrees of freedom can be used in "fundamental tests, ultracold chemistry, and engineering new quantum phases in many-body systems", as argued in Ref. [Mos16].
The diatomic character of the molecules is related to (ultracold) chemistry twofold; on one hand the chemical process of creation of the molecules itself, on the other hand the dimer can be chemically unstable against collisions leading to chemical reactions. The collisional instability depends on the combination of alkali atoms, e.g. KRb molecules [Zuc10]. In this case, the exchange reaction of the type 2 XY → X2 + Y2 is exothermic, i.e. in a collision process two molecules decay into two other homonuclear compounds with more binding energy. The exothermic reaction can be prevented by confining each atom to an individual site of an optical lattice. These processes are but one example of the possibility to explore quantum chemistry with alkali dimers. In the case of RbCs, the absolute ground-state (rovibronic hyperfine ground state) is collisionally stable [Zuc10]. The type of process 2 XY → X2 + Y2 is endothermic and is therefore forbidden. We could already confirm the collisional stability experimentally for the Rb-Cs system [Tak14].
Another advantage of dipolar molecules is that the dipole moment can be arbitrarily "switched" on and off by optical transitions. The electric dipole moment is "switched on" in the ground-state and "switched off" in the excited state. This leads to many possible applications, e.g. for quantum information processing [Yel06]. Summarizing, ultracold alkali dimer systems appear to be all-rounders. The mere challenge of these systems lies in the cooling
down to or close to quantum degeneracy due to the absence of closed cycling transitions. In this thesis I present a generally applicable method to circumvent this obstacle.
1.4 Creation of ultracold dipolar molecular ensembles Several methods exist for creating cold and ultracold dipolar molecules. In this section I
will briefly describe a few of the most promising techniques. The major difference lies within direct cooling of molecules or creating dimers out of ultracold atoms.
Magnetic-field and AC-Stark decelerators only slow packages of molecules, hence the name. But, the possibility to achieve final velocities almost in the range for evaporative cooling in optical dipole traps or magnetic traps [Car09] makes molecular decelerators worth mentioning. These decelerators rely on rapidly varying magnetic or electric fields. The working principle of such decelerators is similar to optical Sisyphus cooling [Dal89; Met99]. First, a molecular source produces packages of molecules by pulsed emission of molecules. These packages enter a region with a very high inhomogeneous magnetic or electric field. The potential energy of the molecules increases in the strong field region by reducing the motional energy. Abruptly switching off the field, when the molecule package encounters the highest field, leads to the maximum possible reduction of kinetic energy. Many subsequent steps of this procedure can efficiently reduce the longitudinal velocity of molecules down to almost stand-still [Hog07; Nar08; Tar04]. For this process it is crucial that the switching of the high magnetic or electric fields is done synchronously with the passing-by of the molecule package. Experiments could achieve slowing of molecular packages with final longitudinal velocities of a few 10 m/s and a final translational temperature in the milli-Kelvin regime was reported [Bet99; Lav11].
Another method is buffer-gas cooling. Here, cryogenic Helium is used to sympathetically cool molecular ensembles down to ∼ 1 K [Wei98]. Collisions between the molecules and the cryogenic Helium are used to reduce the temperature of the molecules. The residual coolant still poses an issue for the cold molecules and ways of removing the coolant were developed. The efforts resulted in setups where a cryogenic buffer gas was used to pre-cool the molecules and a guide led the molecules into a different cooling stage, as described in [Bar11]. Typical temperatures that can be reached are in the milli-Kelvin regime. The temperature gap from 1 K to a few mK, resulting from buffer-gas cooling, can be bridged for example by a conventional magneto-optical trap (MOT) [Met99]. The difficulty lies again within the absence of (optical) cycling transitions, but making use of one or more so-called repumping lasers allows to tailor quasi-closed transitions in a few specific molecules, e.g. SrF and CaF [Ste16; Cha17]. Molecular MOTs are capable of producing samples with a temperature below 1 mK, similar to their atomic siblings, but the final molecular densities (2.5 · 105 cm−3) and phase-space densities (PSDs) (6 · 10−14) are quite low, as the work on SrF shows. The Phase-space density is defined as
PSD = ρλ3 dB = ρ · ( 2π~2
mkBT )
3 2 , (1.7)
with density ρ and de Broglie wavelength λdB, massm, Boltzmann constant kB and temperature T . Note, Bose-Einstein condensation occurs when PSD ≥ 1 [Ket99; Avd06; Wan15].
Magnetic-field decelerators, AC-Stark decelerators and molecular MOTs require additional cooling to reach the regime of quantum degeneracy. Evaporative cooling might be an option but that would require a higher PSD for a sufficient thermalization rate. Additionally, several new aspects have to be considered for the evaporation process like dipolar interactions or long-lived collisional compounds [May13].
A third method and the one we use is the production of molecules out of ultracold atoms. This method has been suggested e.g. by P. Zoller and co-workers [Jak02]. This is the only method so far to my knowledge that is capable of producing dipolar molecules in the absolute rovibronic ground-state with nano-Kelvin temperatures, e.g. KRb [Ni08], RbCs [Tak14; Mol14], NaK [Par15] and NaRb [Guo16]. In all of the mentioned cases, two atomic species are first
1.4 Creation of ultracold dipolar molecular ensembles 7
brought to quantum degeneracy. Second, atoms are associated to weakly-bound molecules by an adiabatic magnetic field sweep across an MFR. Third, a specific two-photon process, called STIRAP (STImulated Raman Adiabatic Passage, cf. Chap. 8.3), transfers the Feshbach molecules into the targeted ground state.
The final yield of ground-state molecules of this method is low, usually several thousands of molecules in total. But this technique is superior to the other techniques mentioned above as it is the only experimentally demonstrated method that can actually produce molecules in the nano-Kelvin regime. It is also superior to other proposals, e.g. atom-by-atom assemblers using additional final transfer [Liu17]. Starting from degenerate samples, associating atoms to molecules by means of a MFR or photoassociation, e.g. in the RbSr case [Bay17], and final STIRAP transfer has lead to the realization of samples with an average entropy per particle as low as ≈ 2.2kB [Mos15b]. This work additionally uses an optical lattice. The bottleneck of this method is the inefficient creation of weakly bound molecules, see e.g. [Tak14]. The limiting effects for the reduced molecule production efficiency were the strong interspecies repulsion an strong losses due to inelastic collisions (cf. Refs. [Pil09a; Ler10; Tak12; Tak14] ).
It is very difficult to produce dipolar ground-state molecules out of atomic ensembles with a PSD close to or even on the order of 1. In many cases of bosonic atomic mixtures, and also in the Rb-Cs case, the inter- and intraspecies scattering properties lead to immiscibility and three-body loss. The Pauli blockade of fermions could resolve these issues, since several decay channels would be forbidden. But the different density profiles of a fermionic cloud and a bosonic cloud are problematic and they have to be matched for efficient molecule production [Mos15b].
1.5 Thesis goals and thesis overview
Po te
n ti
Position x
Po te
n ti
a l V
Figure 1.3.: Illustration of the production route of ground-state molecules that defines the structure of this thesis. First, two spatially separated quantum degenerate samples of Rb and Cs (I) are loaded into an optical lattice, discussed in Chap. 2- 5. We drive Cs into the MI state. The transport of the other SF species towards the immobile MI (II) is discussed in Chap. 6. Chapter 7 explains how the interactions are tuned to zero via an interspecies MFR to create Rb-Cs pairs at each lattice site and how molecules are associated by an magnetic field sweep (III). The molecule production efficiency and possible STIRAP transfer routes are discussed in Chap. 8.
The goal of my thesis is to develop a generally applicable method to overlap spatially- separated quantum-degenerate atomic samples to efficiently produce dipolar ground-state molecules in an optical lattice. Additional to the creation of quantum-degenerate atomic samples (Chap. 2) three ingredients are required for efficient production of a dipolar quantum gas with low entropy :
• tunable interspecies interactions,
• one-atom per site MI,
The final state of our method is an insulating state of dipolar molecules. Note, the previously mentioned techniques, direct cooling of molecules or molecule-production from a trapped gas, lack the advantage of creating a state with isolated molecules on individual lattice wells. Collisional loss of molecules due to the exothermic reaction channel 2 XY → X2 + Y2 is very often necessary. This exothermic reaction channel was investigated in the case of polar KRb [Ni10; Mir11]. Our method can prevent molecule collisions even for an unpolarized sample. The method presented here is not restricted to bosonic quantum gases but is also applicable to molecule production with Bose-Fermi mixtures.
The structure of my thesis follows our sequence for production of ultracold ground-state molecules depicted in Fig. 1.3: Two spatially separated quantum degenerate samples (I) are produced and an optical lattice is turned on (II) to produce a MI of Cs. The SF Rb is transported in the optical lattice and overlapped with Cs. A MFR is used for tuning the interspecies interactions and for production of weakly-bound molecules (III). The Rb-Cs project in Innsbruck has shown previously to efficiently produce two spatially separated BECs of Rb and Cs [Ler11] which is the first building block of our general scheme for ground-state molecule production. The second building block comprises my work, consisting of testing the stability of the BEC production and the realization of the subsequent steps: implementation of an optical lattice, SF transport, mixing of both species and molecule production.
During the implementation and characterization of the optical lattice, we found several technical issues. We had to first modify the setup with focus on the improvement of the optical dipole trap (ODT) setup and magnetic-field setup. I will first review all involved cooling steps for Rb and Cs to reach quantum degeneracy in Chap. 2. Then, I will report on changes to the experimental setup in Chap. 3 and a slight modification of the route for creating two stable BECs of Rb and Cs. I will focus on the improvements of the ODTs and the magnetic-field. These improvements were essential to meet the required stability and performance criteria for highly efficient ground-state molecule production. The tools and methods for finding the stability criteria are discussed in chapters 4 - 7. In particular, the setup and the relevant properties of our simple-cubic (SC) [Kit86] optical lattice are discussed in Chap. 4. The measurements for characterizing the lattice are reported in Chap. 5. The requirements for SF transport of Rb in presence of an optical lattice with a movable ODT and adiabatic merging of both samples are discussed in Chap. 6. In Chap. 7 I will introduce the relevant aspects of magnetic MFRs, which are a core element for merging the Rb and Cs clouds and for producing weakly-bound RbCs molecules, and explain their applications in the Rb-Cs system.
Chapters. 2- 6 introduce all necessary ingredients for our proposed general scheme of efficient production of dipolar ground-state molecules. In Chap. 8 I will explain how to combine these ingredients for highly efficient production of ultracold molecules. I will present our results of producing weakly-bound molecules in the optical lattice. I will also briefly discuss possible STIRAP transfer routes to the ground-state and I will point out a few improvements to increase the overall molecule-production efficiency further.
In conclusion, this thesis will show a novel and highly efficient production scheme for creating a stable ensemble of ultracold rovibronic and hyperfine ground-state molecules. Exploring dipolar systems is feasible with our setup when additionally an external electric field
1.5 Thesis goals and thesis overview 9
is applied for polarizing the sample. This Rb-Cs setup is a powerful system for investigations in many research fields like quantum information processing, quantum chemistry, few- and many-body physics, etc, and is particularly well suited for simulating solid state systems.
2Overview of cooling Rb and Cs
I give a brief overview of the techniques used for production of two spatially separated BECs of Rb and Cs. Double BEC production was already described in detail in Refs. [Ler10; Ler11]. Several parts of the setup were recycled from a previous Cs experiment, described in Refs. [Ryc04; Eng06]. This chapter is an overview of the route to double-BEC production. The path for creating two BECs is illustrated in Fig. 2.1.
Opt. latticeEvaporationRes. down
8000ms 7ms 7ms 15ms 500ms 1000ms 5750ms
Figure 2.1.: Illustration of the timeline for the first part of our general scheme for efficient production of ultracold molecules. It consists of all necessary cooling steps to produce two quantum degenerate samples of Rb and Cs. The duration texp of each involved cooling stage is given.
The protocol for creating two quantum degenerate samples starts with loading of a dual- species magneto-optical trap (MOT) from a dual-species Zeeman slower. After a compressed MOT (cMOT) stage and optical molasses cooling the atoms are further cooled by dual-species Raman sideband cooling (DRSC). We load a large-volume optical dipole trap (ODT), which we call "reservoir" trap. From the reservoir trap the atoms are reloaded during the "Res. hold" and the "Res. down" stage into smaller ODTs ("dimple" traps). The dimples are used for evaporative cooling down to quantum degeneracy. The double BEC is the starting point for my work to efficiently create ultracold molecules in an optical lattice.
2.1 Oven and Zeeman slower We use a double-species oven consisting of two compartments for Rb and Cs, respectively.
The oven design is described in Ref. [Wil09]. The atoms are heated to about 100°C and exit the ovens with a velocity of ∼200 m/s. They pass a differential pumping section [Eng06] before entering an atomic-beam decelerator. This decelerator, based on the Zeeman-slowing technique, reduces the speed of the atoms to meet the capture velocity of our double species magneto-optical trap (MOT) [Met99]. A Zeeman slower uses radiation pressure, exerted by a resonant laser beam. The Doppler shift is compensated by the Zeeman shift that is introduced by a spatially varying magnetic field.
Originally the Zeeman-slower was built for Cs only, but it was adapted to handle Rb and Cs simultaneously by reconfiguring the electric current through the Zeeman slower coils for best slowing conditions for both species. Two independent laser systems for Rb and Cs produce the light for the Zeeman-slower and the MOT [Eng06; Pil05; Pil09b; Pra08; Ler11]. The Rb and Cs laser light for the Zeeman slower are transported by optical fibers to the main table and overlapped at a dichroic mirror before the beams enter the vacuum chamber. Two photodiodes are used for independent monitoring and stabilization of the Rb and Cs Zeeman-slower laser intensities. After the Zeeman slower stage the atoms enter the main glass cell (see Fig. 3.5) where Rb and Cs are captured by a dual species MOT.
2.2 Magnetic fields Our machine uses three pairs of coils to independently create a homogeneous magnetic
field Bhom (coils nearly in Helmholtz configuration) and a gradient field Bgrad (coils nearly in "anti"-Helmholtz configuration). The coil alignment is schematically drawn in Fig. 2.2. The
11
Figure 2.2.: Schematic drawing of the glass cell and main magnetic field coils together with the MOT beams. The atoms are slowed by the Zeeman slower (not shown) and enter the glass cell along the direction as indicated. The atoms are trapped in the center of the glass cell by a MOT. The coil setup for creating the magnetic field consists of six co-axial coils. The three coil pairs are used to independently create a field with a linear gradient (levitation field) and homogeneous field along the vertical (z) axis. The glass cell was re-used from the previous Cs experiment and has a superpolished glass prism underneath [Eng06], that is not used any more. The typical vacuum pressure in the glass cell is ∼ 5 · 10−11 mbar.
coils of each pair are connected in series for best symmetry of the magnetic field. Details on the individual coil pairs are given in Ref. [Pil05; Eng06].
We can choose two configurations: two coil pairs generate the gradient field Bgrad and one coil pair produces the homogeneous field Bhom (initial configuration, used for large Bgrad). Or, one coil pair creates the gradient field Bgrad and two coil pairs are used for the homogeneous field Bhom (new configuration, used for large Bhom). A facility-internal cooling cycle provides the cooling water for the coils. The cooling pipe is directly mounted onto the cooling plate and has the shape of single loop. A detailed description of the configuration can be found in Ref. [Ler10] and in Chap. 3.2, where I will briefly discuss both settings to explain the changes and improvements to the coil setup.
Applications of the levitation and homogeneous fields We use two sets of magnetic coils to create a field to levitate the atoms and to control the
energy levels of the atoms. The energy levels of the hyperfine ground-state of alkali atoms in a magnetic field shift according to the Breit-Rabi Formula [Bre31]:
E(B,mF) = − Ehfs
2I + 1 + x2. (2.1)
The "+"-sign refers to the upper hyperfine states, F Cs = 4 and FRb = 2, and the "-"-sign refers to the lower hyperfine manifolds of F Cs = 3 and FRb = 1. The nuclear Landè factor is gI and the hyperfine Landè factor is gF. The hyperfine splitting is Ehfs, B describes the magnetic field and
x = (gJ − gI)µBB
12 Chapter 2 Overview of cooling Rb and Cs
All relevant parameters and constants for Rb and Cs can be found in [Ste15] and [Ste09].
Levitation field
The levitation field is used to create the linear gradient field for the MOT but also for holding the atoms against gravity (magnetic levitation) at a later stage of the experimental sequence. The magnetic levitation for Rb and Cs has been described in [Pil09b; Ler10], where also the optimum levitation condition for Rb and Cs are discussed. We can balance the gravitational force (Fgrav = mg) by the magnetic force Fmag according to the levitation condition
Fgrav = gFµBmF ∂B
∂z (2.3)
by adapting the value of the magnetic field gradient Bgrad = ∂B/∂z. The magnet force on the atom Fmag = −∇E(B,mF) (cf. Eq. 2.1) can be reduced to Fmag = ∂E(B,mF)/∂z when the atom cloud is located at the vertical symmetry axis of the gradient field.
Homogeneous field
The homogeneous field is used, according to Eq. 2.1 to shift the atomic levels. This, in combination with the coupling of nuclear spin I with the electronic spin S of the form I · S, gives rise to MFRs [Chi10]. The coupling of two alkali atoms is given by the Hamiltonian
~2
2µrm
) + h1 + h2 + V (R) (2.4)
The interatomic distance is R, µrm is the reduced mass, V (R) is the interaction between both atoms and
ha = ζIa · Sa + geµBBSza + gIµBBIza (2.5)
is the monomer Hamiltonian including the Zeeman effect. The coupling strength of the electron spin to the nucleus is given by ζ and ge is the electron Landé factor. A detailed explanation of the coupling for Rb-Cs MFRs at low field can be found in [Tak12] . This coupling is exploited for adjusting the intra- and interspecies scattering properties, interaction tuning and molecule production (Chap. 7).
2.3 Magneto-optical trap A magneto-optical trap (MOT) is a fundamental and well-established tool for producing
ultracold atomic ensembles [Raa87]. The working principle of a MOT is explained in detail in Ref. [Met99]. A MOT uses red-detuned laser light on a closed optical transition, also called cooling transition. The force exerted by the laser light on the atoms is position dependent in presence of a magnetic field gradient. Therefore, pairwise counter-propagating laser beams and a magnetic quadrupole field allow particle trapping and cooling in a well defined spot.
Our 3D MOT uses the levitation field coils. For each direction (x, y, z) we use two identical and counter-propagating MOT laser beams comprising the light for the cooling transition and the "repumper" light for a closed cycling transition. We use the same atomic transitions for the cooling and repumping as described in Ref. [Pil09b]. The MOT-beam configuration is schematically drawn in Fig. 2.2. The Rb and Cs light for the MOT is first overlapped at a dichroic mirror and then sent to the main experiment table by an optical fiber, where it is split into the different MOT beams. The beam size, laser frequency detuning and beam intensities are the same as in Ref. [Pil09b]. Details on our Rb and Cs MOT scheme as well as the Rb and Cs laser system can be found in [Eng06; Pil05; Pil09b; Pra08; Ler11].
We have decommissioned the monitoring of the MOT atom number by the MOT-fluorescence photodiodes, explained in [Pil09b]. Likewise, the MOT-recapture technique got abandoned,
2.3 Magneto-optical trap 13
since we now fully rely on absorption imaging with a CCD camera (Andor iXon DV885-KCS-VP) and automated read-out. The limited space around the glass cell demanded rebuilding and implementation of dichroic mirrors for the MOT setup. This allowed to overlap the ODT beams and the lattice beams with the MOT beams. As a result, the optical lattice beams also intersect at approximately 90° angle. After the MOT has finished loading, the gradient field is ramped up and the laser detuning is increased. This is called "compressed MOT" (cMOT) and increases the PSD. The cMOT is operated for Rb and Cs simultaneously and a subsequent optical molasses cools the atoms further to about 40 µK and 50 µK, respectively. The duration of the cMOT and molasses stage is 15 ms. This leaves us with an optimized system that can reliably produce two cMOTs with an atom number as high as NRb = 2.5 · 107 and NCs = 2.0 · 107 for Rb and Cs respectively. We could reduce the loading time of the MOT to 8 s with respect to [Pil09b].
The compromise between the optimum Rb and Cs settings for the Zeeman slower reduces the loading efficiency of the MOT. We greatly benefit now from increased stability of the MOT and the subsequent cooling steps but the total atom number is decreased with respect to previous work [Pil09b].. Note, both MOTs are slightly displaced from each other to reduce the effect of light assisted collisions [Pil09b].
2.4 Dual species Raman sideband cooling (DRSC) Raman sideband cooling is used to reduce the temperature of atoms [Ker00; Tre01] before
loading them into an ODT. The lowest state of Raman sideband cooling is a dark state and all atoms are pumped into this state where they are not exposed to resonant light any more. Degenerate Raman sideband cooling (DRSC) for two species was first demonstrated with our machine [Pil09b].
The basic concept of DRSC for Rb and Cs is depicted in Fig. 2.3. After the optical molasses cooling, an optical lattice with near-resonant light is switched on and a small homogeneous field along the vertical axis is applied. The atomic population gets distributed among the vibrational states of the lattice. The vertical magnetic field has to be adjusted such that the vibrational levels |νDRSC of the lattice become degenerate with the Zeeman splitting of the magnetic mF-sublevels. The near-resonant lattice light drives Raman transitions between the states |mF, νDRSC and |mF − 1, νDRSC − 1 (double-headed arrows). Laser light, operating at the σ+ transition of the D2 line, resonantly excites the atoms into the upper state from where they spontaneously decay back. This leads effectively to optical pumping back into the states |mRb
F = 1 for Rb and mCs F = 3 for Cs. In this process 2 vibrational quanta were lost. This
leads to a cooling cycle until all atoms accumulate in the lowest state |mRb F = 1, νRb
DRSC = 0 and |mCs
F = 3, νCs DRSC = 0. Atoms can gather during the process in |mRb
F = 0, νRb DRSC = 0
|mCs F = 2, νRb
DRSC = 0 and are kicked back into the cooling cycle by a "polarizer" beam, operating on the π transition of the D2 line. Finally, all atoms are pumped into a dark state (lowest state), where they are "immune" to heating caused by resonant light. Due to the vertical offset field and the optical pumping into the dark state the sample has become polarized. The reduction of vibrational excitations leads to a final temperature of ≈ 3 µK in our case.
The setup of our DRSC system is explained in detail in [Pil09b] but slight adjustments were made: the DRSC laser beams for the near-resonant (dual-color) lattice are first combined at a dichroic mirror and then sent through the same optical fiber to the experiment. Unfortunately, the Rb Raman-lattice laser lacks power and modification of the setup is recommended to increase the DRSC efficiency. We use three DRSC pulses of 3 ms duration. The pulses are separated by 1.5 ms. The magnetic offset field is adjusted for each DRSC pulse individually. This improves the loading efficiency of Rb and Cs into the large volume optical dipole trap ("reservoir" trap). We changed the detuning of the Rb Raman lattice and we use 18 GHz red-detuned light for Rb and 9 GHz red-detuned light for Cs.
π
B
+10-1
σ+
π
σ+
Figure 2.3.: Illustration of dual-species Raman sideband cooling for Rb (red) and Cs (blue). Near- resonant light is used to create an optical lattice. A magnetic field along the z-axis is adjusted such that the Zeeman sublevels are degenerate with the vibrational levels of the lattice. The lattice light drives Raman transitions between the magnetic sublevels and simultaneously lowers the vibrational level. Resonant light on the D2-line(σ+ and π transition) and spontaneous emission are used to pump the atoms into the lowest state (dark state), indicated by the red and blue atoms. Image taken from [Ler10].
2.5 Evaporation Evaporation is used to overcome the limitations of laser cooling by lowering the external
confinement and releasing the hot atoms from the trap while elastic collisions provide re- thermalization [Ket96]. We use optical dipole traps to confine the atoms and we control the collisional properties by controlling the s-wave scattering length as with the homogeneous magnetic field for evaporation.
2.5.1 Optical dipole traps The ODT setup is a core element of our quantum gas apparatus. We perform spatially
separated evaporation of the energetically lowest Rb and Cs spin state in only with the ODTs ("all optical evaporation") without a magnetic trap. The levitation field is only used to aid optimum evaporation conditions by reducing the effect of gravitational sag in the ODTs [Ler10].
A scheme of spatially separated evaporation of Rb and Cs was implemented because simultaneous evaporation in the same ODT has proven to be difficult [Ler11]. The requirement for spatially separated and simultaneous evaporation demanded the realization of movable ODTs. The ODT setup is schematically drawn in Fig. 2.4. The reservoir trap provides sufficient depth and spatial overlap for capturing the large sample after DRSC. Tightly focused ODTs, so-called "dimples", are added. The dimples are used to perform forced evaporation to reach quantum degeneracy. A blue-detuned laser beam for Cs prevents Cs from entering the Rb dimple [Ler11; Ler10] and the spatial separation of the Rb dimple permits only a small fraction of Rb to enter the Cs dimple.
A zoomed-in view will be presented in Fig. 3.1(a), where the difficulties of the old ODT setup and the advantages of the improved new setup are explained (cf. also Chap. 3.1). A more detailed illustration is presented in Fig. 3.5, where the new improved magnetic field
2.5 Evaporation 15
Figure 2.4.: Schematic view of the main chamber. All ODTs use laser light with 1064 nm (red beams). The Rb dimple, where an additional laser beam at 830 nm, overlapped with the Rb dimple, is used for Rb selective trapping (dark red beam). The coordinate axis is identical with Fig. 2.2. The x axis is along the transport direction of Rb. The z-axis is antiparallel to the direction of gravity. For a better view onto the setup, only the coils below the glass cells are shown.
setup is discussed. Note, the ODT beams and the MOT beams incline angles of 5°with the coordinate axis.
2.5.2 Evaporation and Bose-Einstein condensation Thermalization by elastic collisions is essential for cooling down to quantum degeneracy. Rb
was among the first BEC species [And95]. It was doubted that condensation of Cs is possible because of the high rate of spin-relaxation processes in magnetically trappable states [Gué98]. Spin relaxation is one example of inelastic loss. We work with the energetically lowest states of Rb and Cs, so it takes three particles to conserve energy and momentum. Three particles collide and form one dimer molecule and a (free) atom. The binding energy of the molecule is converted into kinetic energy. Two-body loss and collisions with the background gas (one-body loss) can be neglected. Evaporation in optical traps instead of magnetic traps is essential for cooling Cs down to quantum degeneracy and efficient evaporative cooling of Cs in its lowest spin state was demonstrated [Web03a; Kra04]. While (inelastic) three-body collisions must be avoided, elastic collisions are necessary for thermalization of the ensemble. Rb already has a favorable ratio of elastic to inelastic collisions. For Cs a magnetic field region was found around 21 G where the ratio of elastic to inelastic collisions allow efficient evaporation [Kra04]. Combined evaporation of Rb and Cs in the same dipole trap is for the reason of strong three- body losses difficult and different techniques were developed for production of a Rb-Cs double BEC [Ler11; McC11]. We also used a magnetic field of 21 G for simultaneous evaporation of Rb and Cs [Ler10]. An overview on the measured Rb-Cs interspecies loss with our experiment can be found in Ref. [Ler10]. The impact of collisional loss on our general scheme for production of molecules in the optical lattice is discussed in Chap. 7.
The Zeeman slower, MOT, cMOT, optical molasses, DRSC, and all-optical evaporation in dimple traps is the first set of ingredients for our "Rb-Cs quantum optics toolbox" for creating two quantum degenerate samples of Rb and Cs. Each individual cooling stage is done simultaneously for both species. This is the starting point for the work with an optical lattice
for efficient production of ultracold molecules. Before I reveal all details about the lattice, I will continue with explaining the necessary technical improvements and briefly characterize our BECs with our new setup.
2.5.3 The s-wave scattering length as
The interaction between the particles is not only important for evaporation but it also plays a crucial role for molecule creation. The s-wave scattering length as is used to describe the interaction properties of the atoms. I will briefly derive as and define the scattering cross section σ and the interaction parameter g. Details can be found e.g. in Ref. [Dal99] and for the tuning of as with an MFR in Ref. [Chi10].
We start with a potential Vsc(r) with spherical symmetry that scatters off an incoming particle. The scattering particle has mass m1, the scattered particle has mass m2. The scattered, or incoming, particle has momentum k and the involved collisional energy Ecoll is given by Ecoll = (~k)2/(2m2), assuming that the scatterer is at rest in the lab frame. By using the symmetry of this problem and introducing the angle θ between the direction of the incident particle k/k and the line of observation r/r, one can switch to the center-of mass frame. The Hamiltonian is
H = −~ 2µrm
( 1 r2
) + Vsc(r) (2.6)
Here, µrm = m1 ·m2/(m1 +m2) denotes the reduced mass. The solution can be derived with the following ansatz
ψk(r) = eikr + f(θ, k)e ikr
r . (2.7)
The outgoing wave should be expressed in terms of spherical harmonics Ylm(θ) and a phase shift δl. This leads to
f(θ, k)e ikr
2 l + δl
) Ylm(θ). (2.8)
The expansion of the wave function is useful when reducing the problem to the low energy scattering regime. In the low-energy regime the cross section σ(k) is depends on the momentum k. It is energy dependent, since E = ~2k2/(2µrm). The scattering cross section σ(k) can be written as an integral of the scattering amplitude f(θ, k) over the full solid angle:
σ(k) = ∫
4π |f(θ, k)|2 d (2.9)
The total cross section σ(k) can further be expressed in a partial-wave expansion
σl(k) = 4π k2 (2l + 1) sin2 δl, (2.10)
where δl ∝ k2l+1 represents a phase shift that is different for each partial wave l [Sak94]. Since we work in the ultracold regime, it becomes evident to investigate the finding in Eq. 2.10 at ultralow temperatures, meaning that the energy of the incoming particle tends to zero, i.e. k → 0. In the ultracold regime the centrifugal barrier l(l + 1)/(2mr2), which can be on the order of several milli-Kelvin for l = 1, is higher. Thus, this derivation can be simplified by only considering l = 0, i.e. s-wave scattering. Then the cross section becomes independent of the collision energy and takes the much simpler form
σ0(k) = 4πa2 s . (2.11)
2.5 Evaporation 17
The s-wave scattering length in the low-energy limit is defined as
as = − lim k→0
. (2.12)
Detailed knowledge of the long-range part of the van der Waals potential, usually represented by VvdW = C6/r
6, is required for accurate calculation of as. Especially in the Rb-Cs case VvdW supports a very weakly bound state with a binding energy of Eb ≈ 2π~ · 110 kHz, rendering as very large away from MFRs. This is called the background scattering length aBG.
For alkali atoms the absolute value of the s-wave scattering length is on the order of 101 a0 - 102 a0. Comparing this length scale with interatomic distances of a few a0 it becomes legitimate to approximate the scattering potential by a delta-like potential
Vsc(r) = g · δ(r) (2.13)
which must be spherically symmetric again.The scattering potential in Eq. 2.13 describes a contact interaction. The interaction strength depends linearly on the scattering length as and is represented by the parameter
g = 4π~2as
m , (2.14)
with the particle mass m for a single species. The coupling parameter g(1,2) for two different particles is given by
g(1,2) = 2π~2a (1,2) s
µrm , (2.15)
3Improvements of the experimental setup
I will describe the old ODT setup and its limitations first. I will discuss the new setup and explain the improvements. Then I will discuss the changes of the magnetic-field setup. Finally, I will show the adjusted evaporation sequence with the new setup and discuss the quality of the achievable double BECs.
3.1 Optical dipole trap setup 3.1.1 Old dimple trap setup
The full set of ODTs, that is required for simultaneous condensation of Rb and Cs, comprised the large volume ODT (reservoir trap), a shared dimple (SD), a Cs dimple (CsD), a Rb dimple (RbD) and a dimple with 829.5 nm laser light, which was overlapped with the Rb dimple. A schematic overview of the old ODT setup that was installed in the beginning of my thesis is shown in Fig. 3.1. In the first version of the old setup the Cs dimple trap could be moved. This has been described in Ref. [Ler10]. The Rb dimple was aligned at the center of the experiment (cf. Figs 2.4 and 3.2(a)), which is defined as the center of the levitation field. The reservoir-trap beam has a waist of roughly 600 µm and uses up to 70 W of laser power at 1070 nm. The old and the new setup of the reservoir beam is basically the same, it was only slightly realigned during the rebuilding of the MOT beams.
Rb dimple
Reservoir trap
Shared dimple
Cs dimple
x
y
Figure 3.1.: (a) Top view of the old configuration of the ODTs. The image is adopted from Ref. [Ler10]. The Rb dimple consists of two beams, one at 1064 nm (red) and one at 829.5 nm (blue). In this configuration the Cs dimple is movable along one horizontal direction, indicated by the arrow. We control the movement with a modified mirror mount.
We use the SD, the RbD and the CsD and the levitation field for evaporation. All dimple traps are powered by a home-built fiber amplifier at 1064.5 nm [Ler10]. The SD provides confinement along the y-axis. The ODT setup bares the difficulty to prevent spilling of the atoms along the SD trap. Atoms which leave along the SD lead to admixtures of the other species. Investigations on the Rb-Cs evaporation have shown that admixtures would result in three-body loss processes that lead to heating [Ler10]. We can avoid this heating effect by moving the Cs away from the experiment center. This reduces the Rb admixture in the Cs trap and Rb leaves the Cs trap quickly because of the fast Rb-Cs thermalization rate and the smaller trap depth for Rb. Rb takes most of the heat load of the Rb-Cs mixture, leaving a colder Cs sample behind. Thus, a small Rb admixture in the Cs trap is desired [Ler10]. The
19
RbD is overlapped with an 820 nm beam that is blue detuned for Cs and prevents Cs from accumulating in the RbD.
The main ingredient of the old design was a movable optical dipole trap. It consisted of a mirror mount with a piezo actuator. The piezo-actuator (Thorlabs:PZS001) with strain gauge was inserted between the fine-adjustment screw and the mirror-mounting plate. The piezo actuator was installed on the horizontal axis of the mirror mount. An external electronic driver (Thorlabs:AMP001) read out the strain gauge and the signal was fed into an analogue PID controller. The PID controller sent a signal to a high voltage amplifier that served as the piezo driver. The second version of the old ODT setup had a second and identical movable mirror mount to move also the Rb ensemble along the x-direction as well. This setup was able to displace the Cs dimple by more then 400 µm and the Rb dimple by about 130 µm away from the experiment’s center.
We determine the position of the experiment center in a two-step approach. First, for a rough calibration of the center position, we minimize the displacement of the MOT at the MOT’s standard-gradient field and at very high-gradient field, given perfect MOT-beam balancing. When no displacement along both horizontal axis can be detected, we make a CMOT and define the experiment’s center as the center of the CMOT. For the second step, we established a new and much more precise procedure to evaluate the center of the gradient field. We produce a Rb BEC and let it expand without any confinement but perfectly hold it against gravity with the levitation field. Rb is preferred due to its larger magnetic moment and smaller mass. The Rb atoms are in the center of the levitation field when no residual magnetic force pulls them away. We had to install additional coils("tracking coils") to cancel residual magnetic forces by displacing the levitation field (cf. Chap. 3.2.2).
The old dimple trap setup was unstable. The instabilities were observed e.g. by producing a Cs BEC, holding it against gravity with the levitation field and then switching off the dipole traps. We observed after ttof = 50 ms of time of flight (TOF) position fluctuations of the BEC of up to 50 µm. We analyzed if the electronic noise in the circuit of the dimple piezos is to blame for the position instabilities. We could rule that out because the position uncertainty resulting from the electronic noise is < 1 µm. We found that two other sources were the main reason for this problem: non-laminar air-flow around the glass cell and insufficient mechanical stability of the dimple-beam setup. This manifests itself in beam-pointing instabilities of the dimple traps and results in a small velocity in a random direction of the BEC after release from the ODTs.
3.1.2 New dimple trap setup We use the same set of ODTs in the new ODT setup and the same 1064.5 nm laser source but
the opto-mechanical setup was changed to improve the beam-pointing stability of all dimple traps.
The beam launch for the SD points upwards, allowing for an installation very close to the glass cell. A λ/2-waveplate and a polarizing beam splitter are used for defining the polarization. We use four lenses to produce a beam with wSD = 39.0(5) µm radius at the position of the atoms. We used trap-frequency measurement method described in Ref. [Ler10] to obtain the values for the beam waist. The setup of the SD is schematically drawn in Fig. 3.2(b). A 45°-tilted mirror deflects the beam and steers it into the glass cell.
The design of the combined RbD and CsD (Design and setup by T. Takekoshi) aims at best mechanical stability while still allowing for the translation of the Rb and Cs dimple. The new design allows to move the Rb dimple along the horizontal and vertical axis and the Cs dimple along the vertical axis. This is indicated by the arrows in Fig. 3.2(a). The beam waist of the Cs dimple can be reduced from about 122 µm to about 61 µm, indicated by the ellipses in Fig. 3.2(a). This allows to bring Rb closer to Cs after condensation without spilling Cs into the Rb trap and essentially reduce the travel distance of Rb in the optical lattice. Both degrees of freedom of each dimple trap are provided by moving only the fiber heads in front
20 Chapter 3 Improvements of the experimental setup
of the fiber-collimator lens instead of using a steering mirror. Limited space required us to sacrifice the horizontal movement of the Cs dimple in exchange for vertical movement because the periscope (mirrors M2 and M3) was designed to guide the beam in one plane. Now the periscope guides the beams upwards so the upper beams have 90° angle with respect to the lower beams.
Figure 3.2.: (a) Schematic drawing of the new Shared-dimple, Rb-dimple and Cs-dimple beam alignment. The arrows at the end of the RbD and CsD indicate the travel direction of the traps and the ellipse indicate the variable beam waist of the CsD. (b) Schematic drawing of the assembly of the optical components of the SD in a lens tube (Thorlabs: SM1 Lens Tube system). Instead of using our standard (small) molded aspheric lens (Thorlabs TME220-C), we chose an aspheric lens with a larger NA of 0.54 (Thorlabs AL2520C, focal length fL1 = +20 mm). The rotatable PBS defines the polarization and is optimized for maximum transmission. A telescope, consisting of lenses L2 (Casix PCV100, fL2 = −100 mm ) and L3 (Casix PCX125, fL3 = +125 mm ), adjusts the beam diameter for lens L4 (Casix PCX300, fL4 = 300 mm ), which focuses the beam down to a desired waist of wSD = 39.0(5) µm. (c) Schematic drawing of the new combined Rb dimple and Cs dimple beam launch. The fibers adapters FA1 and FA2 are mounted onto two two-axis flexure stages. The optical fibers OF1 (Nufern, PM980-HP) and OF2 (Nufern, PM780-HP) can be rotated for optimum transmission through the polarization-cleaning PBSs. PBS1 is used for polarization cleaning of the Cs dimple beam, PBS2 and PBS3 clean the polarization of the 829.5 nm dimple and the Rb dimple. The collimation lenses F1 and F2 are from Thorlabs (TME220-C, fL1 = fL2 = 11 mm). The dimple beams are overlapped at PBS4. The telescope consists of lenses L3 (Thorlabs LC2679-B, fL3 = +60 mm), L4 (Thorlabs AC254-060 B, fL4 = +60 mm) and L5 (Thorlabs AC254-300 B, fL4 = +300 mm). A periscope, M2 and M3, is used to steer the dimple-trap beams into the glass cell.
The optical components of the new Rb dimple and Cs dimple are shown in Fig. 3.2(c). Both fiber adapters FA1 and FA2 are mounted on individual flexure stages. The flexure stage for the CsD can move left-right and forward-backward, as indicated by the arrows. This results in a vertical movement of the beam and adjustable beam waist (cf. also Fig. 3.3). The RbD flexure stage can move left-right and up-down allowing horizontal and vertical movement of the RbD. Both fibers OF1 and OF2 can be rotated to achieve maximum transmission through the polarization cleaning PBS1, PBS2 and PBS3. The Cs dimple beam is overlapped with the Rb dimple beam at PBS4. The subsequent telescope produces a beam waist of wCs = 122(2) µm and wCs = 55(1) µm for the CsD and RbD, respectively. The beam waist was measured with the trap-frequency measurement method described for our system in Ref. [Ler10] with an uncertainty of about 2%. We detected small alignment imperfections of FA1 with respect to lens L1 and FA2 with respect to L2 which leads to small cross-talk of the travel directions. This means that moving a dimple beam in one direction results in a tiny change of the other degree of freedom. We find this is a minor issue and we did not improve it since we could not find evidence that it limits the evaporation, and we can compensate it during Rb transport and the production of RbCs molecules.
3.1 Optical dipole trap setup 21
All dimple beams and all optical-lattice beams use AOMs for intensity stabilization. All AOM frequencies fAOM were the same in the old setup. We changed the relative detunings frel = fAOM,1 − fAOM,2 to avoid interference between the beams. The current values for fAOM, frel and the polarization of the beams are shown in Tab. 3.1
SD Rb dimple Cs dimple Lattice x Lattice y Lattice z fAOM (MHz) +105.1 -109.7 115.1 -115 +110 -105 Polarization v h v h h h frel (Pol.) SD Rb dimple Cs dimple Lattice x Lattice y Lattice z (MHz) SD – 215 (⊥) 225 (⊥) 5 (⊥) 220 (⊥) 5 (⊥) Rb dimple 5 (⊥) – 225 (⊥) 5 () 220 () 5 ( / ⊥) Cs dimple 10 () 225 (⊥) – 230 (⊥) 5 (⊥) 220 (⊥) Lattice x 215 (⊥) 5 (⊥) 230 (⊥) – 225 (⊥) 10 ( / ⊥) Lattice y 5 (⊥) 220 () 5 (⊥) 225 (⊥) – 215 ( / ⊥) Lattice z 210 (⊥) 5 ( / ⊥) 220 (⊥) 10 ( / ⊥) 215 ( / ⊥) –
Table 3.1.: Polarization and AOM frequencies fAOM used for intensity stabilization of the optical lattice and dimple traps. The polarization is distinguished between in-plane (h) and out-of-plane (v) polarization. The plane is defined as the horizontal (x−y) plane. The relative frequency frel = fAOM,1 − fAOM,2 gets defined by the intensity-stabilizing AOMs and is the frequency difference of the trapping beams. The and ⊥ signs indicate the polarization setting of the beams with respect to each other.
Figure 3.3.: 3D CAD illustration of the new combined dimple launch in Fig. 3.2(c). A few suspension elements were hidden to enable better view onto the relevant optical and optomechanical parts. The black arrows on the lower left corner of the piezo stages indicate the travel directions of the fiber mounts. The overlayed red line and the dashed turquoise line represents the beam path of the 1064.5 nm and 829.5 nm light, respectively. The optical elements are labeled according to Fig. 3.2
A 3D CAD drawing of the new Rb and Cs dimple setup is shown in Fig. 3.3. The optical
fibers OF1 and OF2 are mounted onto the flexure stages. This arrangement magnifies the travel range of the beam from 100 µm of the piezo flexure stage to 1.4 mm at the position of the atoms. Black arrows indicate the travel direction of the flexure stages. We pick up the light that is reflected from PBS2 to 4 for stabilizing the intensity of the 829.5 nm dimple, the Rb-dimple and the Cs-dimple beam, respectively. The beam path of the 829.5 nm dimple has a tiny, but negligible deviation from the Rb dimple because we use angle polished fiber connectors (Thorlabs FC/APC). We decided to use the same optical fiber for transporting the Rb dimple light and the 829.5 nm light to main chamber as in the old setup [Ler10] .
The position instabilities of the BECs after ttof = 50 ms were reduced to a few µm with the new dimple trap setup. This was an important step for the work with the optical lattice.
3.2 Magnetic-field setup 3.2.1 Main magnetic-field coils
The initial configuration used one homogeneous-field coil pair and two levitation-field coil pairs [Eng06; Pil09b]. The dimensions of the coil setup deviate significantly from the optimum values for Helmholtz configuration (homogeneous field coils) and anti-Helmholtz configuration (levitation field) [Eng06]. A schematic drawing of the coil setup is shown in Fig. 3.4. This configuration was capable of producing a magnetic field of Bhom < 230 G. We use an H-bridge for switching the quadrupole field on and off and to control its direction. The quadrupole coils are used for operating the MOT and to levitate the atoms (cf. Chap. 2.2). The total magnetic field noise of this setup was roughly 10 mG when a lead acid battery was used instead of the power supply (Delta Elektronika SM 30-200). The field noise was measured with a calibrated pick-up coil that had been used for magnetic field modulation spectroscopy [Tak12].
Cooling pipe
Glass cell
CB
A
A
Figure 3.4.: Schematic drawing of the main magnetic field setup, which consists of three pairs of coils (A,B,C). Two pairs (A,B) are designed with 25 windings each (large rectangles) and one pair (C) has 10 windings each (small squares). The yellow-colored coils (C) are used for producing the homogeneous magnetic field Bhom and the green-colored coils (B) are used for producing the levitation field Bgrad. We can change the configuration and use one coil pair (A) either for the homogeneous field of for the levitation field. Figure adapted from [Ler10].
During my thesis it became clear that we have to access higher magnetic field regions around 350 G. We had to rule out the option to run more current through the homogeneous field coil pair C by either stacking power supplies or use a more powerful supply or stack lead acid batteries. We do not have enough cooling power and the electronic driver stage is insufficient for any of these options. Another possibility is to use two coil pairs for the homogeneous field (coil pairs A and C in Fig. 3.4). We also benefit from the fact that the magnetic moments of Rb as well as Cs increase with increasing field strength Bhom [Ste15; Ste09; Ler10] and therefore
3.2 Magnetic-field setup 23
one levitation field coil is sufficient. A 3D view of the main magnetic coil system with the glass cell and the dimple beams is presented in Fig. 3.5.
Figure 3.5.: Sketch of the glass cell assembly with the surrounding magnetic coils. The "tracking coils 1" create a homogeneous field that is −45°-rotated with respect to the x-axis in the horizontal plane. The "tracking coils 2" are perpendicular to the first ones and create another homogeneous field in the horizontal plane. The tracking coils are mounted with aluminum profiles on the cooling plate. The coil pairs A, B and C are mounted onto the cooling plate. Water is used as coolant and the flux through the cooling pipe is roughly 33ml/s.
We decided to replace the old current drivers for the homogeneous field coils and the levitation field coils because of insufficient power handling. The design goal for the new magnetic field system was ≈ 5 ·10−5 stability and high magnetic field ramp speed. The stability criterion can be derived from the properties of the MFR that we use for interaction tuning and molecule production (cf. Chap. 7). The maximum field and the width of the smallest relevant MFR give an upper limit for the magnetic field instabilities: the smallest accessible MFR has a width of 50 mG at 197 G and the maximum magnetic field is ≈ 670 G. This leads to a requirement for the magnetic field stability of better than 50 · 10−3/670 ≈ 1 · 10−4. The stabilization scheme for the magnetic field is the same as in Ref. [Eng06]. We have measured a maximum magnetic field ramp speed of 0.6(2) G/µs. This is the slope dB/dt that we measured with a linear ramp from minimum to maximum field in the shortest amount of time. The total magnetic-field noise is roughly 50 mG. The new electronic driving circuit for the entire coil system is described in Chap. A.
3.2.2 Tracking-coil system On top of the random movement of the BECs that was caused by the unstable ODTs also a
constant force was pulling the atoms away from the center of the experiment. We have found evidence that the levitation field coils and the homogeneous field coils are slightly misaligned. Additionally, magnetized objects around the glass cell caused a constant magnetic force on the atoms and we replaced as many as possible.
We observed the effect of the residual magnetic force at at the end of the evaporation. Atoms were spilled out of the dimple traps and we observed that the BECs were accelerated during TOF. The small misalignment of the magnetic field coils, tilted or uncentered with respect to each other, caused the additional magnetic gradient that lead to a tilt of the dimple traps. The magnetic force also caused heating (see Chap. 4.5.2) when the optical lattice was on.
The following experiment was used to quantify the effect of the magnetic force on a Rb or Cs BEC. We produced a Cs BEC and move it to the center of the experiment. Note that this measurement was done after the dimple trap setup was improved. We observed that the BEC was accelerated to the side and the acceleration was depending on the strength of the homogeneous field Bhom. The position of the Cs BEC as a function of the expansion time ttof is shown in Fig. 3.6. We fit a parabola to the data and extract the acceleration of the BEC. In the case for low field the BEC is accelerated with 3.4(1) m/s and at high field with -0.40(5) m/s. We also measured the magnetization of magnetized objects around the glass cell with the "Fluxmaster" of Stefan Mayer Instruments and found that they produced a magnetic field bigger than 20 mG.
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 4
0
4
8
1 2
1 6
2 0 B h o m = 2 1 G B h o m = 2 1 0 G
Dis tan
ce to
ce nte
(µm )
E x p a n s i o n t i m e t t o f ( m s )
Figure 3.6.: Horizontal distance dCs to the center of the experiment of the Cs BEC after release from the ODT. The homogeneous field is Bhom = 21 G and the levitation field is Bgrad = 31.1 G/cm for the first two measurements and Bhom = 210 G and Bgrad = 27.2 G/cm for the third measurement. The solid lines are parabolic fits to the data. The acceleration is 3.4(1) m/s for the measurements at low field and -0.40(5) m/s for the measurement at high field.
We cancel the unwanted magnetic field gradient by adding a small homogeneous magnetic field Btrac along the horizontal plane. The additional horizontal-field co

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