We consider a mixture of single-component bosonic and fermionic atoms with an interspecies interaction that is varied using a Feshbach resonance. By performing a mean-field analysis of a two-channel model, which describes both narrow and broad Feshbach resonances, we find an unexpectedly rich phase diagram at zero temperature: Bose-condensed and non-Bose-condensed phases form a variety of phase-separated states that are accompanied by both critical and tricritical points. We discuss the implications of our results for the experimentally observed collapse of Bose-Fermi mixtures on the attractive side of the Feshbach resonance, and we make predictions for future experiments on Bose-Fermi mixtures close to a Feshbach resonance.
Gross-Pitaevskii equation with gain is used to model Bose Einstein condensation (BEC) fed by the surrounding thermal cloud. It is shown that the number of atoms continuously injected into BEC from the reservoir can be controlled by applying the external magnetic field via Feshbach resonance.
We consider the dissociation of a molecular Bose-Einstein condensate during a magnetic-field sweep through a Feshbach resonance that starts on the molecular side of the resonance and ends on the atomic side. In particular, we determine the energy distribution of the atoms produced after the sweep. We find that the shape of the energy distribution strongly depends on the rate of the magnetic-field sweep, in a manner that is in good agreement with recent experiments.
We discuss the atom-atom scattering problem across a Feshbach resonance in a two-dimensional dilute Bose gas at zero temperature, in the limit where the s-wave scattering length exceeds the width of the vertical confinement. We determine a tunable coupling-strength parameter and by controlling it we evaluate how the condensate wave function spreads out with increasing atom-atom repulsions. We also discuss the stability of the condensate in the magnetic-field regime where the coupling has become attractive.
The production of pairs of fast atoms leads to a pronounced loss of atoms during upward ramps of Feshbach resonance levels in dilute Bose-Einstein condensates. We provide comparative studies on the formation of these bursts of atoms containing the physical predictions of several theoretical approaches at different levels of approximation. We show that despite their very different description of the microscopic binary physics during the passage of a Feshbach resonance, all approaches lead to virtually the same prediction on the total loss of condensate atoms, provided that the ramp of the magne…
We present how to control interactions between solitons, either bright or dark, in Bose-Einstein condensates by synchronizing Feshbach resonance and harmonic trap. Our results show that as long as the scattering length is to be modulated in time via a changing magnetic field near the Feshbach resonance, and the harmonic trapping frequencies are also modulated in time, exact solutions of the one-dimensional nonlinear Schrödinger equation can be found in a general closed form, and interactions between two solitons are modulated in detail in currently experimental conditions. We also propose expe…
Complex, localized stable solitons, characterized by a power law behaviour, are found for a quasi-one-dimensional Bose-Einstein condensate near Feshbach resonance. Both dark and bright solitons can be excited in the experimentally allowed parameter domain, when two and three-body interactions are respectively repulsive and attractive. These solutions are obtained for non-zero chemical potential, unlike their unstable real counterparts which exist in the limit of vanishing $μ$. The dark solitons travel with constant speed, which is quite different from the Lieb mode, where profiles with differe…
In this paper, we study the three-body recombination rate of a homogeneous dilute Bose gas with a Feshbach resonance at zero temperature. The ground state and excitations of this system are obtained. The three-body recombination in the ground state is due to the break-up of an atom pair in the quantum depletion and the formation of a molecule by an atom from the broken pair and an atom from the condensate. The rate of this process is in good agreement with the experiment on $^{23}$Na in a wide range of magnetic fields.
We put forward a scheme for controlling Josephson-like tunneling in two-component Bose-Einstein condensates coupled with microwave field via Feshbach resonance and tuning aspect ratio of trapping potential. We prove how to realize a perfect periodic oscillation from a fast damped and irregular oscillation on relative number of atoms in future experiment. In particular, intensity of Josephson-like tunneling can be successfully controlled through controlling speed of recovering the initial value of intra-atomic interaction and aspect ratio of trapping potential. Interestingly, we find that relat…
We predict the resonance enhanced magnetic field dependence of atom-dimer relaxation and three-body recombination rates in a $^{87}$Rb Bose-Einstein condensate (BEC) close to 1007 G. Our exact treatments of three-particle scattering explicitly include the dependence of the interactions on the atomic Zeeman levels. The Feshbach resonance distorts the entire diatomic energy spectrum causing interferences in both loss phenomena. Our two independent experiments confirm the predicted recombination loss over a range of rate constants that spans four orders of magnitude.
We measure the effect of a magnetic Feshbach resonance (FR) on the rate and light-induced frequency shift of a photoassociation resonance in ultracold $^7$Li. The photoassociation-induced loss rate coefficient, $K_p$, depends strongly on magnetic field, varying by more than a factor of 10$^4$ for fields near the FR. At sufficiently high laser intensities, $K_p$ for a thermal gas decreases with increasing intensity, while saturation is observed for the first time in a Bose-Einstein condensate. The frequency shift is also strongly field-dependent and exhibits an anomalous blue-shift for fields j…
We give an overview of recent experiments on an ultracold Fermi-Bose quantum gas where the interspecies interaction can be tuned via magnetic Feshbach resonances. We first describe the various steps that have led to the observation of Feshbach resonances in the K-Rb system we investigate, and their accurate characterization. We then describe experiments in which Feshbach resonances are exploited to study interaction effects and to associate weakly bound KRb dimers.
We consider an atomic Fermi gas confined in a uniform optical lattice potential, where the atoms can pair into molecules via a magnetic field controlled narrow Feshbach resonance. The phase diagram of the resulting atom-molecule mixture in chemical and thermal equilibrium is determined numerically in the absence of interactions under the constraint of particle conservation. In the limiting cases of vanishing or large lattice depth we derive simple analytical results for important thermodynamic quantities. One such quantity is the dissociation energy, defined as the detuning of the molecular en…
In this work we revisit the topic of two-dimensional Bose-Einstein condensates under the influence of time-dependent magnetic confinement and time-dependent scattering length. A moment approach reduces the examination of moments of the wavefunction (in particular, of its width) to an Ermakov-Pinney (EP) ordinary differential equation (ODE). We use the well-known structure of the solutions of this nonlinear ODE to ``engineer'' trapping and interatomic interaction conditions that lead to condensates dispersing, breathing or even collapsing. The advantage of the approach is that it is fully tract…
We study the steady state of a zero-temperature Bose gas near a Feshbach or photoassociation resonance using a two-channel mean-field model that incorporates atomic and molecular condensates, as well as correlated atom pairs originating from dissociation of molecules into pairs of atoms. We start from a many-body Hamiltonian for atom-molecule conversion, and derive the time dependent version of the mean-field theory. The stationary solution of the time dependent model is rendered unique with an approximation that entails that all noncondensate atoms are correlated, as if emerging from dissocia…
We consider a mixture of two-species atomic Bose-Einstein codensates coupled to a bound molecular state at zero temperature via interspecies Feshbach resonance. The interspecies Feshbach coupling precludes the possibility of doubly mixed phases while enables not only the pure molecular superfluid but also the pure atomic superfluids to exist as distinct phases. We show that this system is able to support a rich set of phase separations, including that between two distinct mixed atom-molecule phases. We pay particular attention to the effects of the Feschbach coupling and the particle collision…
In many experiments involving conversion of quantum degenerate atomic gases into molecular dimers via a Feshbach resonance, an external magnetic field is linearly swept from above the resonance to below resonance. In the adiabatic limit, the fraction of atoms converted into molecules is independent of the functional form of the sweep and is predicted to be 100%. However, for non-adiabatic sweeps through resonance, Landau-Zener theory predicts that a linear sweep will result in a negligible production of molecules. Here we employ a genetic algorithm to determine the functional time dependence o…
We test a non equilibrium approach to study the behavior of a Bose-Fermi mixture of alkali atoms in the presence of a Feshbach resonance between bosons and fermions. To this end we derive the Hartree-Fock-Bogoliubov (HFB) equations of motion for, the interacting system. This approach has proven very successful in the study of resonant systems composed of Bose particles and Fermi particles. However, when applied to a Bose-Fermi mixture, the HFB theory fails to identify even the correct binding energy of molecules in the appropriate limit. Through a more rigorous analysis we are able to ascribe …
We study suppression of the collapse and stabilization of matter-wave solitons by means of time-periodic modulation of the effective nonlinearity, using the nonpolynomial Schroedinger equation (NPSE) for BEC trapped in a tight cigar-shaped potential. By means of systematic simulations, a stability region is identified in the plane of the modulation amplitude and frequency. In the low-frequency regime, solitons feature chaotic evolution, although they remain robust objects.
We study a dilute mixture of degenerate bosons and fermions across a Feshbach resonance of the Fermi-Fermi scattering length $a_F$. This scattering length is renormalized by the boson-induced interaction between fermions and its value is crucial to determine the phase diagram of the system. For the mixture in a box and a positive Bose-Fermi scattering length, we show that there are three possibilities: a single uniform mixed phase, a purely fermionic phase coexisting with a mixed phase, and a purely fermionic phase coexisting with a purely bosonic one. As $1/a_F$ is increased from a negative v…
We use a Feshbach resonance to tune the scattering length a of a Bose-Einstein condensate of 7Li in the |F = 1, m_F = 1> state. Using the spatial extent of the trapped condensate we extract a over a range spanning 7 decades from small attractive interactions to extremely strong repulsive interactions. The shallow zero-crossing in the wing of the Feshbach resonance enables the determination of a as small as 0.01 Bohr radii. In this regime, evidence of the weak anisotropic magnetic dipole interaction is obtained by comparison with different trap geometries.
We report on the creation of heterospecies bosonic molecules, associated from an ultracold Bose-Bose mixture of 41K and 87Rb, by using a resonantly modulated magnetic field close to two Feshbach resonances. We measure the binding energy of the weakly bound molecular states versus the Feshbach field and compare our results to theoretical predictions. We observe the broadening and asymmetry of the association spectrum due to thermal distribution of the atoms, and a frequency shift occurring when the binding energy depends nonlinearly on the Feshbach field. A simple model is developed to quantita…
We study the bosonic atoms with a wide Feshbach resonance at zero temperature in terms of the renormalization group. We indicate that this system will always collapse in the dilute limit. On the side with a positive scattering length, the atomic superfluid is an unstable local minimum in the dilute limit and it determines the thermodynamics of this system within its lifetime. We calculate the equilibrium properties at zero temperature in the unitary regime. They exhibit universal scaling forms in the dilute limit due to the presence of a nontrivial zero temperature, zero density fixed point. M…
We investigate the stability properties of breather solitons in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length and con ned only by a one dimensional optical lattice. We compare regions of stability in parameter space obtained from a fully 3D analysis with those from a quasi two-dimensional treatment. For moderate con nement we discover a new island of stability in the 3D case, not present in the quasi 2D treatment. Stable solutions from this region have nontrivial dynamics in the lattice direction, hence they describe fully 3D breather …
We demonstrate a reversible conversion of a Li2 molecular Bose-Einstein condensate to a degenerate Fermi gas of atoms by adiabatically crossing a Feshbach resonance. By optical in situ imaging, we observe a smooth change of the cloud size in the crossover regime. On the Feshbach resonance, the ensemble is strongly interacting and the measured cloud size is 75(7)% of the one of a non-interacting zero-temperature Fermi gas. The high condensate fraction of more than 90% and the adiabatic crossover suggest our Fermi gas to be cold enough to form a superfluid.
We present a theoretical model for Bragg scattering from a Bose-Einstein condensate (BEC) in the vicinity of a magnetic Feshbach resonance, using a two c-field formalism, one c-field for the atom and the other for a molecule formed of two atoms. We use this model to numerically simulate a recent experiment [S. B. Papp et al., Phys. Rev. Lett., 101(13):135301, Sep 2008] investigating the effects of strong interactions on the Bragg spectrum from a 85Rb BEC. Results from these simulations are in very good quantitative agreement with the experimental results, confirming the importance of the reson…
Feshbach resonances are the essential tool to control the interaction between atoms in ultracold quantum gases. They have found numerous experimental applications, opening up the way to important breakthroughs. This Review broadly covers the phenomenon of Feshbach resonances in ultracold gases and their main applications. This includes the theoretical background and models for the description of Feshbach resonances, the experimental methods to find and characterize the resonances, a discussion of the main properties of resonances in various atomic species and mixed atomic species systems, and …
We analyze gap solitons in trapped Bose-Einstein condensates (BECs) in optical lattice potentials under Feshbach resonance management. Starting with an averaged Gross-Pitaevsky (GP) equation with a periodic potential, we employ an envelope wave approximation to derive coupled-mode equations describing the slow BEC dynamics in the first spectral gap of the optical lattice. We construct exact analytical formulas describing gap soliton solutions and examine their spectral stability using the Chebyshev interpolation method. We show that these gap solitons are unstable far from the threshold of loc…
In this paper, we study the phase diagram of a homogeneous Bose gas with a repulsive interaction near a wide Feshbach resonance at zero temperature. The Bose-Einstein-condensation (BEC) state of atoms is a metastable state. When the scattering length $a$ exceeds a critical value depending on the atom density $n$, $na^3>0.035$, the molecular excitation energy is imaginary and the atomic BEC state is dynamically unstable against molecule formation. The BEC state of diatomic molecules has lower energy, where the atomic excitation is gapped and the molecular excitation is gapless. However when the…
Using a Feshbach resonance, we create ultracold fermionic molecules starting from a Bose-Fermi atom gas mixture. The resulting mixture of atoms and weakly bound molecules provides a rich system for studying few-body collisions because of the variety of atomic collision partners for molecules; either bosonic, fermionic, or distinguishable atoms. Inelastic loss of the molecules near the Feshbach resonance is dramatically affected by the quantum statistics of the colliding particles and the scattering length. In particular, we observe a molecule lifetime as long as 100 ms near the Feshbach resona…
Sound waves are observed and studied in an optically trapped degenerate Fermi gas of spin-up and spin-down atoms with magnetically tunable interactions. Measurements are made throughout the crossover region, from a weakly-interacting Fermi gas through the resonant Fermi superfluid regime to a Bose condensate of dimer molecules. The measured sound velocities test the equation of state and confirm the universal hypothesis.
We demonstrate a novel method of inducing an optical Feshbach resonance based on a coherent free-bound stimulated Raman transition. In our experiment atoms in a Rb87 Bose-Einstein condensate are exposed to two phase-locked Raman laser beams which couple pairs of colliding atoms to a molecular ground state. By controlling the power and relative detuning of the two laser beams, we can change the atomic scattering length considerably. The dependence of scattering length on these parameters is studied experimentally and modelled theoretically.
In this paper we formulate the time-dependent many-body theory of photoassociation in an atomic Bose-Einstein condensate with realistic interatomic interactions, using and comparing two approximations: the first-order cumulant approximation, originally developed by Kohler and Burnett [Phys. Rev. A 65, 033601 (2002)], and the reduced pair wave approximation, based on a previous paper [Phys. Rev. A 68 033612 (2003)] generalizing to two channels the Cherny-Shanenko approach [Phys. Rev. E 62, 1046 (2000)]. The two approximations differ only by the way a pair of condensate atoms is influenced by th…
We consider an atomic Fermi gas confined in a uniform optical lattice potential, where the atoms can pair into molecules via a magnetic field controlled narrow Feshbach resonance. Thus by adjusting the magnetic field the portion of fermionic and bosonic particles in the system can be continuously varied. We analyze the statistical mechanics of this system and consider the interplay of the lattice physics with the atom-molecule conversion. We study the entropic behavior of the system and characterize the temperature changes that occur during adiabatic ramps across the Feshbach resonance. We sho…
A sum rule approach is used to calculate the zero temperature oscillation frequencies of a two component trapped atomic Fermi gas in the BCS-Bose Einstein condensation crossover region. These sum rules are evaluated using a local density approximation which explicitly includes Feshbach molecules. Breathing modes show non-monotonic behavior as a function of the interaction strength, while quadrupole modes are insensitive to interactions for both spherically symmetric and axially symmetric traps. Quantitative agreement is found with experiments on atomic $^6Li$ system and with other theoretical …
We derive pseudo-potentials that describe the scattering between two particles in two spatial dimensions for any partial wave m, whose scattering strength is parameterized in terms of the m-dependent phase shift. Using our m=0 pseudo-potential, we develop a coupled channel model with 2D zero-range interactions, which describes the two-body physics across a Feshbach resonance. Our model predicts the scattering length, the binding energy and the "closed channel molecular fraction" of two particles; these observables can be measured in experiments on ultracold quasi-2D atomic Bose and Fermi gases…
We calculate the critical temperature for Bose-Einstein condensation in a gas of bosonic atoms across a Feshbach resonance, and show how medium effects at negative scattering lengths give rise to pairs reminiscent of the ones responsible for fermionic superfluidity. We find that the formation of pairs leads to a large suppression of the critical temperature. Within the formalism developed by Nozieres and Schmitt-Rink the gas appears mechanically stable throughout the entire crossover region, but when interactions between pairs are taken into account we show that the gas becomes unstable close …
We explore the Bose condensed phases of an atomic gas on the molecular side of a Feshbach resonance. In the absence of atom-molecule and molecule-molecule scattering, we show that the atomic condensate is either a saddle point of the free energy with energy always exceeding that of the molecular condensate, or has a negative compressibility (hence unstable to density fluctuations). Therefore no phase transition occurs between the atomic and molecular condensates. We also show that a repulsive molecule-molecule scattering can stabilize a sufficiently dense atomic condensate, leading to the poss…
A universal characterization of interactions in few- and many-body quantum systems is often possible without detailed description of the interaction potential, and has become a defacto assumption for cold atom research. Universality in this context is defined as the validity to fully characterize the system in terms of two-body scattering length. We discuss universality in the following three contexts: closed-channel dominated Feshbach resonance, Efimov physics near Feshbach resonances, and corrections to the mean field energy of Bose-Einstein condensates with large scattering lengths. Novel e…
Mean-field approach has recently been used to model coupled atom-molecular Bose-Einstein condensates (BEC) and coupled Fermi-Bose condensates near Feshbach resonance. Sweeping of magnetic field across the resonance gives a new (nonlinear) version of Landau-Zener problem. We investigate the structure of the corresponding classical phase space and calculate change in the action which corresponds to finite-rate efficiency of the sweep. We consider the case of non-zero initial action, which corresponds to some finite initial molecular fraction.
Within the framework of the variational approach the ground state is studied in a gas of Fermi atoms near the Feshbach resonance at negative scattering length. The structure of the originating superfluid state is formed by two coherently bound subsystems. One subsystem is that of quasi molecules in the closed channel and the other is a system of pairs of atoms in the open channel. The set of equations derived allows us to describe the properties of the ground state at an arbitrary magnitude of the parameters. In particular, it allows one to find a gap in the spectrum of single-particle Fermi e…
The route toward a Bose-Einstein condensate of dipolar molecules requires the ability to efficiently associate dimers of different chemical species and transfer them to the stable rovibrational ground state. Here, we report on recent spectroscopic measurements of two weakly bound molecular levels and newly observed narrow d-wave Feshbach resonances. The data are used to improve the collisional model for the Bose-Bose mixture 41K87Rb, among the most promising candidates to create a molecular dipolar BEC.
We theoretically examine a system of Fermi degenerate atoms coupled to bosonic molecules by a Feshbach resonance, focusing on the superfluid transition to a molecular Bose-Einstein condensate dressed by Cooper pairs of atoms. This problem raises an interest because it is unclear at present whether bimodal density distributions observed recently in 40-K and 6-Li are due to a condensate of bosonic molecules or fermionic atom pairs. As opposed to 40-K, we find that any measurable fraction of above-threshold bosonic molecules are necessarily absent for the 6-Li system in question, which strongly i…
We determine the adiabatic phase diagrams for a resonantly-coupled system of Fermi atoms and Bose molecules confined in a harmonic trap by using the local density approximation. The key idea of our work is conservation of entropy through the adiabatic process. We also calculate the molecular conversion efficiency as a function of the initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field.
A model of two 1D ideal Bose gases A and B with strong AB attractions induced by a p-wave AB Feshbach resonance is studied. The model is solved exactly by a Bose-Bose duality mapping, and it is shown that there is no A-component or B-component Bose-Einstein condensation and no AB-pair off-diagonal long-range order (ODLRO), but both AA-pair and BB-pair ODLRO. After generalization by adding even-wave AA and BB repulsion and reducing the strength of the odd-wave AB attraction by Feshbach resonance detuning, a quantum phase transition occurs between a phase with AB contact nodes and one with no su…
We produce Bose-Einstein condensates of two different species, $^{87}$Rb and $^{41}$K, in an optical dipole trap in proximity of interspecies Feshbach resonances. We discover and characterize two Feshbach resonances, located around 35 and 79 G, by observing the three-body losses and the elastic cross-section. The narrower resonance is exploited to create a double species condensate with tunable interactions. Our system opens the way to the exploration of double species Mott insulators and, more in general, of the quantum phase diagram of the two species Bose-Hubbard model.
We investigate unconventional superfluidity in a gas of Fermi atoms with an anisotropic p-wave Feshbach resonance. Including the p-wave Feshbach resonance as well as the associated three kinds of quasi-molecules with finite orbital angular momenta $L_z=\pm1,0$, we calculate the transition temperature of the superfluid phase. As one passes through the p-wave Feshbach resonance, we find the usual BCS-BEC crossover phenomenon. The p-wave BCS state continuously changes into the BEC of bound molecules with L=1. Our calculation includes the effect of fluctuations associated with Cooper-pairs and mol…
Magnetically tunable Feshbach resonances were employed to associate cold diatomic molecules in a series of experiments involving both atomic Bose as well as two spin component Fermi gases. This review illustrates theoretical concepts of both the particular nature of the highly excited Feshbach molecules produced and the techniques for their association from unbound atom pairs. Coupled channels theory provides the rigorous formulation of the microscopic physics of Feshbach resonances in cold gases. Concepts of dressed versus bare energy states, universal properties of Feshbach molecules, as wel…
We consider the stability of an ultracold trapped Bose-Einstein condensate near a Feshbach resonance. Using a modified Gross-Pitaevskii equation that includes higher-order terms and a multi-channel model of Feshbach resonances, we find regions where stability can be enhanced or suppressed around experimentally measured resonances. We suggest a number of ways to probe the stability diagram. Using scattering length zero-crossings huge deviations are founds for the critical particle number. Effects are enhanced for narrow resonances or tighter traps. Macroscopic tunneling of the condensate is ano…
We suggest a time-dependent dynamical mean-field-hydrodynamic model for the collapse of a trapped boson-fermion condensate and perform numerical simulation based on it to understand some aspects of the experiment by Modugno et al. [Science 297, 2240 (2002)] on the collapse of the fermionic condensate in the 40K-87Rb mixture. We show that the mean-field model explains the formation of a stationary boson-fermion condensate at zero temparature with relative sizes compatible with experiment. This model is also found to yield a faithful representation of the collapse dynamics in qualitative agreeme…
We demonstrate optical tuning of the scattering length in a Bose-Einstein condensate as predicted by Fedichev {\em et al.} [Phys. Rev. Lett. {\bf 77}, 2913 (1996)]. In our experiment atoms in a $^{87}$Rb condensate are exposed to laser light which is tuned close to the transition frequency to an excited molecular state. By controlling the power and detuning of the laser beam we can change the atomic scattering length over a wide range. In view of laser-driven atomic losses we use Bragg spectroscopy as a fast method to measure the scattering length of the atoms.
Scattering length, which can be measured in Bose-Einstein condensate and Feshbach molecule experiments, is extremely sensitive to the variation of fundamental constants, in particular, the electron-to-proton mass ratio (m_e/m_p or m_e/Lambda_{QCD}, where Lambda_{QCD} is the QCD scale). Based on single- and two-channel scattering model, we show how the variation of the mass ratio propagates to the scattering length. Our results suggest that variation of m_e/m_p on the level of 10^{-11}~10^{-14} can be detected near a narrow magnetic or an optical Feshbach resonance by monitoring the scattering …
This PhD tutorial article is a review of our experiments on heteronuclear quantum gas mixtures at the University of Hamburg. We introduce basic properties of trapped Fermi-Bose mixtures and demonstrate the achievement of large quantum degenerate mixtures of 40K and 87Rb. Using heteronuclear Feshbach resonances, we show how the heteronuclear interaction can be tuned, allowing us to induce phase separation and collapse for large repulsive and attractive interactions, respectively. We realize Fermi-Bose mixtures in 3D optical lattices as a novel quantum many-body system and study coherence proper…
