We construct a simple quantum optics style mean field model to investigate the behaviour of a zero-temperature untrapped quantum degenerate Bose gas close to a Feshbach resonance. The model allows for both atomic and molecular condensates as well as correlated zero-momentum “BCS” pairs whose provenance would be dissociated zero momentum molecules. Beginning with a second quantized (momentum representation) Hamiltonian and equations conserving total (free and bound) atom number and enforcing an assumption that atoms only appear either in the condensates or pairs and the usual Bogoliubov approximation, the system is numerically and in certain limits, analytically, soluble in the steady state and exhibits a second order phase transition to a pure atomic condensate when the controllable parameters of the coupling Rabi frequency and detuning are varied across an (analytically determined) transition line. Analysis of the thermodynamics of the zero-entropy system shows a negative pressure and hence mechanical instability on both sides of the resonance. A mathematical difficulty arising from an ultra-violet divergence due to the assumption of a zero range interaction is resolved with the help of a simpler, exactly analytically soluble two atom version of the problem. ^
We construct a simple quantum optics style mean field model to investigate the behaviour of a zero-temperature untrapped quantum degenerate Bose gas close to a Feshbach resonance. The model allows for both atomic and molecular condensates as well as correlated zero-momentum “BCS” pairs whose provenance would be dissociated zero momentum molecules. Beginning with a second quantized (momentum representation) Hamiltonian and equations conserving total (free and bound) atom number and enforcing an assumption that atoms only appear either in the condensates or pairs and the usual Bogoliubov approximation, the system is numerically and in certain limits, analytically, soluble in the steady state and exhibits a second order phase transition to a pure atomic condensate when the controllable parameters of the coupling Rabi frequency and detuning are varied across an (analytically determined) transition line. Analysis of the thermodynamics of the zero-entropy system shows a negative pressure and hence mechanical instability on both sides of the resonance. A mathematical difficulty arising from an ultra-violet divergence due to the assumption of a zero range interaction is resolved with the help of a simpler, exactly analytically soluble two atom version of the problem. ^
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 dissociation of molecules. The steady state is solved numerically, but limiting cases are also found analytically. The system has a phase transition in which the atomic condensate emerges in a nonanalytic fashion. We quantify the scaling of the observable quantities, such as fractions of atomic and molecular condensates, with the detuning and the atom-molecule conversion strength. Qualitatively, the dependence on detuning rounds out with increasing coupling strength. A study of the thermodynamics shows that the pressure of the atom-molecule system is negative, even on the molecule side of the resonance. This indicates the possibility of mechanical instability.
This doctoral dissertation is concerned with the physics of strongly interacting cold alkali atoms at low temperatures near a Feshbach resonance. In Chapter 1, we establish a connection between superfluid 4He and the BCS theory of superconductivity, and cold alkali atoms. We give the history of cold atoms, describing the significant achievements, pitfalls and challenges. In Chapter 2, we explore the thermodynamics of strongly interacting Bosonic atoms. We explore the stability of atomic Bosonic condensates near a Feshbach resonance. We show that the experimentally attained atomic condensate is a saddle point of the free energy, but the kinetics of its decay is slow. We also show that there is a second, higher density condensate branch which has an Ising-like phase transition to a molecular (paired) condensate when ramped across the Feshbach resonance. We argue that due to the high density, inelastic 3-body processes possibly render this transition unobservable. In Chapter 3, we explore the thermodynamics of Fermionic atoms near a Feshbach resonance. We determine the zero-temperature (T less than/less than T sub(F) ) pair propagator for a spinimbalanced mixture of up and down spin Fermions, and use it to show that such a mixture becomes completely polarized at mu sub(down arrow)= -0.9 mu sub(up arrow) . We also determine the Thouless criterion for superfluidity in a spin-imbalanced Fermi mixture, and construct a phase diagram of such a system at zero temperature. We then compare our results with experiments performed by two different groups. We find that interaction modifications to the minority spin self-energy inferred from our analysis is roughly double those observed in experiments. This discrepancy is consistent with the expected accuracy of the theory. In Chapter 4, we extend our analysis of the preceding chapter to calculate the surface tension of an interface between spin-polarized Fermions in the normal and superfluid phases. We show that, as expected, this surface tension decreases with increasing temperature and vanishes at a tricritical temperature, above which the transition becomes continuous. We also calculate the thickness of the interface; at T = 0, this is a few interparticle spacings, but diverges at the tricritical temperature. To compare with a set of relevant experiments, we also develop a phenomenological model for surface tension, and conclude that experimental surface tensions are an order of magnitude higher than what our microscopic calculation yields. We hypothesize possible mechanisms. In Chapter 5, we calculate the finite temperature phase diagram of a Bose-Fermi mixture produced from a spin-imbalanced two-component Fermi gas deep in the BEC phase. We show that there is a discontinuous transition between the superfluid and normal phase, with an entropy of mixing sufficient to cool the system down. We detail the construction of such a cooling scheme to cool a Fermi system below what is possible evaporatively, and find that the cooling efficiency is comparable to typical evaporative schemes. In Chapter 6, we shift our focus from thermodynamics to dynamics. We calculate shifts in the energy spectrum of a spin-balanced Fermionic superfluid of Cooper pairs due to the presence of energetically close states coupled by a Feshbach resonance. These shifts manifest themselves as clock-shifts in the radio-frequency spectrum of the superfluid. In addition to a broad asymmetric peak coming from the break-up of Cooper pairs, we find (for certain parameter ranges) a sharp, symmetric "bound-bound" spectral line coming from the conversion of Cooper pairs in one channel to pairs or molecules in another channel. Our theory shows remarkable quantitative agreement with experiments performed by an experimental group.
We have observed condensation of fermionic atom pairs in the BCS-BEC crossover regime. A trapped gas of fermionic 40K atoms is evaporatively cooled to quantum degeneracy and then a magnetic-field Feshbach resonance is used to control the atom-atom interactions. The location of this resonance is precisely determined from low-density measurements of molecule dissociation. In order to search for condensation on either side of the resonance, we introduce a technique that pairwise projects fermionic atoms onto molecules; this enables us to measure the momentum distribution of fermionic atom pairs. The transition to condensation of fermionic atom pairs is mapped out as a function of the initial atom gas temperature T compared to the Fermi temperature T(F) for magnetic-field detunings on both the BCS and BEC sides of the resonance.
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 equilibria 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 energy spectrum with respect to the atomic one for which half of the atoms have been converted into dimers. Importantly we find that the dissociation energy has a nonmonotonic dependence on lattice depth.
Recent repoet on the formation of two-dimensional Bose-Einstein condensates (BECs) of spinning g-wave molecules is surprise. Here we study quantum phase transition in the quasi-2D atomic Bose gas with a g-wave Feshbach resonance, and show that there are two phase transitions in this system: from a phase with only a atomic Bose-Einstein condensate to a phase with coexistence of atomic and molecular condensation and from the coexistence phase to a phase with only a molecular Bose-Einstein condensate. We show that the g-wave resonance Feshbach has two features in the phase transitions, a phase with only a atomic Bose-Einstein condensate is no longer forbidden and resonance effects appear only at levels beyond the mean-field. We determine the T=0 beyond mean-field phase diagram of the gas as a function of magnetic field and molecular condensation density. We also detemine the ratio of atomic to molecular condensation in the coexistence phase, which can be tested in experiments.
Bose-Einstein condensate (BEC) is considered under conditions of Feshbach resonance in two-atom collisions due to a coupling of atomic pair and resonant molecular states. The association of condensate atoms can form a molecular BEC, and the molecules can dissociate to pairs of entangled atoms in two-mode squeezed states. Both entanglement and squeezing can be applied to quantum measurements and information processing. The processes in the atom-molecule quantum gas are analyzed using two theoretical approaches. The mean-field one takes into account deactivating collisions of resonant molecules with other atoms and molecules, neglecting quantum fluctuations. This method allows analysis of inhomogeneous systems, such as expanding BEC. The non-mean-field approach -- the parametric approximation -- takes into account both deactivation and quantum fluctuations. This method allows determination of optimal conditions for formation of molecular BEC and describes Bose-enhanced dissociation of molecular BEC, as well as entanglement and squeezing of the non-condensate atoms.