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arXiv:0809.1675 · Physical Review A

Functional renormalization for trion formation in ultracold fermion gases

Stefan Floerchinger, Richard Schmidt, Sergej Moroz, CHRISTOF WETTERICH

Published Jan 6, 2009• 46 citations
arXivHTMLPDF

Abstract

The energy spectrum for three species of identical fermionic atoms close to a Feshbach resonance is computed by use of a nonperturbative flow equation. Already a simple truncation shows that for large scattering length $\ensuremath{\mid}a\ensuremath{\mid}$ the lowest energy state is a ``trion'' (or trimer) bound state of three atoms. At the location of the resonance, for $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$, we find an infinite set of trimer bound states, with exponentially decreasing binding energy. This feature was pointed out by Efimov [V. Efimov, Phys. Lett. 33B, 563 (1970)]. It arises from limit cycle scaling, which also leads to a periodic dependence of the three body scattering coupling on $\mathrm{ln}\ensuremath{\mid}a\ensuremath{\mid}$. Extending our findings by continuity to nonzero density and temperature we find that a ``trion phase'' separates a Bose-Einstein condensate (BEC) and a BCS phase, with interesting quantum phase transitions for $T=0$.

DOI
10.1103/physreva.79.013603

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The energy spectrum for three species of identical fermionic atoms close to a Feshbach resonance is computed by use of a nonperturbative flow equation. Already a simple truncation shows that for large scattering length \ensuremath∣a\ensuremath∣\ensuremath{\mid}a\ensuremath{\mid}\ensuremath∣a\ensuremath∣ the lowest energy state is a trion'' (or trimer) bound state of three atoms. At the location of the resonance, for $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$, we find an infinite set of trimer bound states, with exponentially decreasing binding energy. This feature was pointed out by Efimov [V. Efimov, Phys. Lett. 33B, 563 (1970)]. It arises from limit cycle scaling, which also leads to a periodic dependence of the three body scattering coupling on $\mathrm{ln}\ensuremath{\mid}a\ensuremath{\mid}$. Extending our findings by continuity to nonzero density and temperature we find that a trion phase'' separates a Bose-Einstein condensate (BEC) and a BCS phase, with interesting quantum phase transitions for T=0T=0T=0.
The method of functional renormalization is applied to the theoretical investigation of ultracold quantum gases. Flow equations are derived for a Bose gas with approximately pointlike interaction, for a Fermi gas with two (hyperfine) spin components in the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover and for a Fermi gas with three components. The solution of the flow equations determine the properties of these systems both in the few-body regime and in thermal equilibrium. For the Bose gas this covers the quantum phase diagram, the condensate and superfluid fraction, the critical temperature, the correlation length, the specific heat or sound propagation. The properties are discussed both for three and two spatial dimensions. The discussion of the Fermi gas in the BCS-BEC crossover concentrates on the effect of particle-hole fluctuations but addresses the complete phase diagram. For the three component fermions, the flow equations in the few-body regime show a limit-cycle scaling and the Efimov tower of three-body bound states. Applied to the case of Lithium they explain recently observed three-body loss features. Extending the calculations by continuity to nonzero density, it is found that a new trion phase separates a BCS and a BEC phase for three component fermions close to a common resonance. More formal is the derivation of a new exact flow equation for scale dependent composite operators. This equation allows for example a better treatment of bound states.
A functional renormalization-group study for the Bose-Einstein condensate (BEC)-BCS crossover for ultracold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality for broad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentration akFa{k}_{\mathrm{F}}akF​, besides their dependence on the temperature in units of the Fermi energy. In particular, we compute the universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate to which level of accuracy universality holds for gases of Li and K atoms.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
We study the quantum three-body free-space problem of two same-spin-state fermions of mass mmm interacting with a different particle of mass MMM, on an infinitely narrow Feshbach resonance with infinite sss-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio $\ensuremath{\alpha}=m/M$. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within each sector of fixed total angular momentum lll. For $\ensuremath{\alpha}$ increasing from 0, we find that the trimer states first appear at the lll-dependent Efimovian threshold ${\ensuremath{\alpha}}_{c}^{(l)}$, where the Efimov exponent sss vanishes, and that the entire trimer spectrum (starting from the ground trimer state) is geometric for $\ensuremath{\alpha}$ tending to ${\ensuremath{\alpha}}_{c}^{(l)}$ from above, with a global energy scale that has a finite and nonzero limit. For further increasing values of $\ensuremath{\alpha}$, the least bound trimer states still form a geometric spectrum, with an energy ratio $\mathrm{exp}(2\ensuremath{\pi}/|s|)$ that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.
We study ultracold gases of alkali-metal atoms in the quantum degenerate regime. The interatomic interactions in these type of systems can be tuned using resonances induced by magnetic or electric fields. The tunability of the interactions, together with the possibility of confining the atoms with several kinds of external potentials, allows for a completely novel approach to study basic problems in many-body physics, and moreover, allows to enter regimes which have never been accessible in condensed matter or nuclear physics. For example, this has led to the experimental demonstration of an intimate relation between two types of superfluidity: the crossover from Bose-Einstein condensation of tightly bound molecules to the superfluid behavior related to weakly bound BCS-like pairs. Another experimental landmark was reached when the existence of universal Efimov three-body bound states was proven in experiments with ultracold bosonic cesium atoms. In this thesis, we study several aspects of these strongly interacting and ultracold atomic gases. We develop an analytical model that encapsulates all of the relevant scattering physics in atomic systems where open-channel shape resonances and closedchannel Feshbach resonances give rise to complicated and non-trivial scattering properties. This model provides lots of physical insight and is shown to describe important quantities, such as the molecular energies and scattering phase shifts, with a high level of accuracy. The model is compared to full numerical coupled-channel calculations in two atomic systems: rubidium and lithium. We study the BCS-BEC crossover using a many-body description of the ultracold gas that includes the non-trivial energy dependence of the scattering model. We show that it gives rise to superfluid behavior associated with the formation of BCS-like pairs while the low-energy interactions are repulsive in character. The energy dependence of the interactions is crucial, as it gives rise to attractive interactions at the Fermi energy, necessary for the formation of Cooper pairs. We demonstrate new ways of controlling the interatomic interactions using a combination of magnetic and electric fields. This leads to experimental control of, for instance, the three-body parameter in the context of Efimov physics and of non-universal behavior in the BCS-BEC crossover in fermionic gases. Using a four-body method based on first principles, we solve the molecule-molecule scattering problem to calculate several important properties of bosonic and fermionic Summary 145 molecules that consist of light and heavy atoms. These type of molecules are of current experimental interest, and we predict several exciting relations between three- and four-body observables in these type of systems.