A dilute Bose-Einstein condensate (BEC) near a Feshbach resonance provides experimental physics with a clean and controllable system to investigate strongly interacting many-body systems. The ability to tune the scattering length allows BECs to be projected onto strong interactions from an initial weakly interacting state. However, historically, studying a bulk 3D strongly interacting BEC has been diffcult, as these systems are inherently unstable due to three-body inelastic collisions. Thus, 2D, 1D, lattice confined, and two component Fermi gases were used to explore the strong interactions in an ultracold gas. In this thesis, I present the first measurement of a strongly interacting 3D 85Rb BEC. I introduce our experimental system and the techniques we used to probe the BEC. I first report on probes of a BEC in the weakly interacting regime. To probe a spherical 85Rb BEC with strong interactions we implemented a novel experimental technique that allowed us to quickly change the interactions, bypassing inelastic losses. We projected the BEC onto unitarity, where the scattering length diverges and the interactions are infinite, in order to observe dynamics and the unexpectedly long lifetime of the gas. Additionally, we observe a universality of the gas with respect to the length scale set by the interparticle spacing.
This thesis discusses a series of studies that investigate the effects of interaction – essentially the s-wave scattering – in the various properties of Bose-Einstein condensates (BEC). The phonon wavefunction in a BEC was measured using Bragg spectroscopy and compared with the well-known Bogoliubov theory. Phonons were first excited in a BEC of 3× 10 condensed Na atoms via small-angle two-photon Bragg scattering. Large angle Bragg scattering was then used to probe the momentum distribution. We found reasonable agreement with the theory. With the same technique of Bragg diffraction, we studied the four-wave mixing process for matter waves. The BEC was split into two strong source waves and a weak seed wave. The s-wave scattering coherently mixed pairs of atoms from the sources into the seed and its conjugate wave, creating a pair-correlated atomic beams with “squeezed” number difference. A Feshbach resonance was used to produce ultracold Na2 molecules with initial phase-space density in excess of 20. Starting from an atomic BEC, a magnetic field ramp shifted a bound state from above the threshold of the unbound continuum to below, creating a molecular population with almost zero center-of-mass motion. A reverse field ramp dissociated the cold molecules into free atom pairs carrying kinetic energy dependent on the ramp speed. This dependence provided a measure of the coupling strength between the bound state and the continuum. Condensates were loaded into optical lattices formed with retro-reflected single frequency lasers. Quantum phase transition from the superfluid state to Mott-insulator state was observed in a three dimensional lattice. The increased interaction and flattened dispersion relation led to strongly enhanced quantum depletion in the superfluid state. Thesis Supervisor: Wolfgang Ketterle Title: John D. MacAurthur Professor of Physics
This thesis presents experiments probing physics in the crossover between Bose-Einstein condensation (BEC) and BCS superconductivity using an ultracold gas of atomic fermions. Scattering resonances in these ultracold gases (known as Feshbach resonances) provide the unique ability to tune the fermion-fermion interactions. The work presented here pioneered the use of fermionic Feshbach resonances as a highly controllable and tunable system ideal for studying the cusp of the BCS-BEC crossover problem. Here pairs of fermionic atoms have some properties of diatomic molecules and some properties of Cooper pairs. I present studies of a normal Fermi gas at a Feshbach resonance and the work required to cool the gas to temperatures where superfluidity in the crossover is predicted. These studies culminated in our observation of a phase transition at the cusp of the BCS-BEC crossover through condensation of fermionic atom pairs. I also discuss subsequent work that confirmed the crossover nature of the pairs in these condensates.
The year 2015 marks the 20th anniversary of the achievement of Bose–Einstein condensation (BEC) in atomic vapors. Atomic BEC, realized by Eric Cornell, Carl Wieman and Wolfgang Ketterle in 1995, commenced an exciting wave to explore quantum few- and many-body physics. Different from solid state matter and helium superfluids, atomic condensates in the gas phase come with drastically lower densities and temperatures, which place the systems in a unique regime where the atomic interactions are simple and precisely known. Furthermore, the condensates are typically weakly interacting: the interaction length scale of few nanometers is much shorter than the interatomic separation of hundreds of nanometers. Thus many early experimental observations can be well understood based on mean-field theory. One celebrated model is the Gross-Pitaevskii equation which provides a precise and quantitative description of weakly interacting Bose condensates [1]. Opportunities to extend cold atom research beyond the weak interaction regime emerged in the early 2000s. Several experimental schemes were developed to prepare atoms in the strong correlation and interaction regimes. One approach is loading atoms into an optical lattice, the interference pattern of laser beams. The lattice potential enhances the interactions of localized atoms and quenches their kinetic energy. A variety of lattice configurations are realized that promise novel quantum phases of ultracold atoms. Another route to strong interaction is based on Feshbach resonance, typically induced by magnetically shifting a molecular state near the scattering continuum of atoms. The resonant coupling between scattering atoms and the molecular state leads to a divergence of the atomic interaction length scale. Exactly on Feshbach resonance, the collision cross section reaches the quantum mechanical limit, and the system becomes strongly hydrodynamic with mean-free path as short as interparticle separation. Fascinating observations in the strong coupling regime soon followed: prominent examples include the superfluid-Mott insulator transition in optical lattices [2], resonantly interacting Fermi gases in the BEC–BCS (Bardeen–Cooper–Schieffer superfluid) crossover [3] and Efimov three-body states [4], and they lead to tremendous research interest into novel quantum states and quantum dynamics. One of the key motivations in these explorations is to uncover new, universal principles that underpin the physics of strongly correlated quantum systems. A well-studied universal behavior in many-body physics manifests in the critical phenomena near a continuous phase transition. Near the critical point, the system develops a diverging correlation length, which suppresses the dependence on other microscopic length scales. The systems are thus scale invariant and can be universally characterized by a few critical exponents. In cold atoms, divergence of correlation lengthes has been observed near the superfluid phase transition [5,6], as well as the superfluid-Mott insulator quantum phase transition in optical lattices. In the lattice experiments, scale invariance can be tested based on the equation of state measurements [7]. These works complement the transport measurements in condensed matter experiments. Has a new class of universal physics been discovered in atomic quantum gases? Yes, it turns out that a special class of strongly interacting quantum gas with ‘intrinsic’ scaling symmetry emerges near a Feshbach resonance [8]. Near the resonance, the scattering length scale diverges a → ∞ [9], unimaginable in classical systems, and the interaction length reaches the maximum value allowed by quantum mechanics as 1/k, called the ‘unitarity limit’ and k is the momentum of the scattering state. A famous example is the strongly interacting Fermi gas in the BEC–BCS crossover regime, where the resonant interactions lead to strong pairing and and universal hydrodynamic behavior. An exciting development in the study of the BEC–BCS crossover focuses on the Fermi gas exactly on Feshbach resonance. Here the system, also called a ‘unitary Fermi gas’, is the most complex and strongly correlated. The resonant interaction, however, implies a special scaling symmetry of the system. To understand the symmetry, we consider a scale transformation: x → Λx that stretches the coordinates of all atoms by a factor of Λ, and thus reduces the atomic density n by a factor of Λ3 in a 3D gas and the kinetic energy Ek ∝ −∇2 by Λ2. One can see that both kinetic energy Ek and interaction energy in the unitarity limit na → n/k are reduced by the same factor Λ2. The system is thus hypothesized to be invariant under the scale transformation [8]. Remarkably, similar scaling symmetry also holds for 2D quantum gas with a constant coupling g = const. [10]. Here stretching the size of the 2D gas by a factor of Λ also reduces both the kinetic Ek and interaction energy gσ by Λ2, where σ is the 2D density. Since the scaling symmetry in these systems comes directly from the form of the interactions, intrinsic scale invariance is robust at all temperatures T and chemical potentials μ and does not rely on the proximity to a phase transition. The scaling symmetry suggests that the only independent variables are the ratios of the length scales or the equivalent energy scales. In a grand canonical ensemble, the independent ratio can be chosen as μ/kBT. This realization greatly simplifies theoretical description as well as the experimental characterization of the system. Theoretically, a new set of universal thermodynamic relations are expected to be insensitive to the microscopic details; experimentally, measurements are expected to collapse into universal functions that only depend on μ/kBT. Measurements on unitary Fermi gas and the 2D Bose gas are consistent with the scaling hypothesis, shown in Fig. 1(a) and (b). Remarkable quantitative agreement between measurements and calculations has greatly advanced our understanding of the ‘unitary Fermi gas’ and 2D atomic gas. Ultracold atomic gases which are intrinsically scale invariant: (a) unitary Fermi gas, (b) 2D Bose gas and (c) unitary Bose gas. The scaling symmetry suggests that thermodynamic observables only depend on the ratio of chemical potential and thermal energy |
μ~=μ/kBT|. In (a), thermodynamic pressure of a unitary Fermi gas is compared with calculations and other measurements. Figure courtesy of M. Zwierlein [11]. In (b), the collapse of the density measurement at various temperatures (inset) confirms the scaling symmetry. Figure adopted from [12]. Panel (c) shows the three-body loss spectrum in a 6Li133Cs mixture. The geometric progression of the Efimov resonance positions at B = B1, B2 and B3 toward the Feshbach resonance at B0 results from the discrete scaling symmetry of Efimov physics. Figure adopted from [13]. We may expect the same scaling symmetry for a resonantly interacting Bose gas. Here, however, the scaling symmetry is more subtle because of the presence of three-body physics. Predicted by Vitaly Efimov in the context of nuclear physics [14], an infinite number of three-body bound states exist for resonantly interacting bosons and their energies follow a discrete scaling law, breaking the otherwise continuous scaling symmetry. Identical fermions do not form such three-body states due to Pauli's exclusion principle. Theoretically the three-body bound states are supported by the ‘Efimov potential’ V(R)∝ − 1/R2, where R is the characteristic length scale of the three interacting atoms. While the Efimov potential does not break the continuous scaling symmetry, the presence of discrete three-body states does break the symmetry. An alternative picture based on renormalization group limited cycles [15] also explains the discrete scaling symmetry of the system, a scenario that rarely happens in electronic systems. Experimentally, Efimov states have been observed in cold gases of alkali atoms, as well as in helium-4. The geometric scaling symmetry of Efimov states is reported in a recent experiment, shown in Fig. 1(c). The same discrete scaling symmetry is expected for unitary quantum gas of bosons, which, if confirmed, will establish a unique and fascinating class of strongly interacting quantum systems. While in this article, we focus on atoms in optical lattices and unitary atomic gases as primary examples of universal physics that have been actively investigated, efforts in many new research frontiers have shown promising prospects to uncover new universal physics in the strongly correlated and interacting systems. Non-equilibrium dynamics is one of the hottest directions. Ultracold atoms are particularly suited for the study of quantum dynamics due to their slow evolution time scales of miniseconds to seconds. Dynamics near a quantum phase transition and the thermalization of isolated quantum systems are two research topics with many unanswered questions. Cold atom experiments may provide new universal insights that go beyond current theoretical framework. Another exciting direction is quantum many-body systems with long range or very strong interactions. Examples include trapped ions, and condensates of exotic atoms with huge magnetic dipole moments, polar molecules, as well as those with Rydberg excitations. These systems possess various long range interactions. Novel universal dynamics, for example, propagation of information and entanglement, are predicted to depend solely on the asymptotic form of the interaction and the dimension of the systems. The author thanks Yijun Wang for discussion and careful reading of the article. This work was supported by National Science Foundation (PHY-0747907) and Army Research Office-Multiple University Research Initiative (W911NF-14-1-0003).
Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent T-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.
The field of Bose-Einstein condensation (BEC) in dilute atomic gases provides a fruitful playground to test well-developed theories of quantum fluids. Research using BECs can address open questions relating to the many-body aspects of two-component quantum liquids, namely the interaction between the hydrodynamic normal and the superfluid component at finite temperatures. After the first realization of BEC some pilot experiments have been carried out, but detailed experiments are missing. This has to be compared to the case of liquid helium II, where many experiments since the 1950s have added to our understanding of novel phenomena in quantum liquids, like collective excitations, first and second sound, and others. One of the drawbacks of liquid helium is that the interactions are so strong that a clear distinction between the two components is difficult and the interpretation of the phenomena remains ambiguous. Most experimental studies of dilute Bose-Einstein condensed gases use clouds, which are in the collisionless regime. The reason for the lack of detailed experiments in BECs to study quantum liquids and in particular the hydrodynamical aspects of it, is the limited number of atoms (typically 1--10 million) in the experiments leaving the thermal atoms virtually collisionless. Efforts to decrease the mean free path by increasing the confinement limits the lifetime of the sample, since the density is limited by three-body decay. Experiments on dilute clouds of cold atoms are generally conducted in highly asymmetric traps. In these elongated, cigar-shaped geometries the mean free path of the atoms can become much shorter than the size of the cloud in the long, axial direction, but at the same time exceeds the size in the other, radial directions. In this so-called hydrodynamic-collisionless regime the system can be considered radially collisionless and axially hydrodynamic. In our setup we have created BECs containing up to 300 million sodium atoms by evaporation of atoms in an axially strongly decompressed trap with an aspect ratio of 1:65. Hot atoms created in three-body collisions are able to leave the sample in this highly asymmetric trap, before they can heat other atoms in an avalanche. The sample is axially hydrodynamic, but due to the large aspect ratio collisionless in the radial direction. Such samples are ideal for the observation of the interaction between the superfluid and normal fluid component of the cloud. In this thesis we describe the setup which is used to reach the hydrodynamic regime and we describe a novel implementation of the well-known phase contrast imaging technique. Next, we study collective excitations in the crossover from the collisionless toward the hydrodynamic regime, we determine the heat conduction in a cold, hydrodynamic thermal cloud and observe the propagation of a thermal wave in the two-fluid regime. Furthermore, we study the friction between the superfluid and normal fluid component of the cloud. In the final chapters of this thesis we study the propagation of second sound, as well as the formation and propagation of shock waves.