Measurement of the entropy and critical temperature of a strongly interacting Fermi gas

Spin Susceptibility above the Superfluid Onset in Ultracold Fermi Gases

Ultracold atomic Fermi gases can be tuned to interact strongly, which produces a display of spectroscopic signatures above the superfluid transition reminiscent of the pseudogap in cuprates. However, the extent of the analogy can be questioned since many thermodynamic quantities in the low temperature spin-imbalanced normal state can be described successfully using Fermi liquid theory. Here we present spin susceptibility measurements across the interaction strength-temperature phase diagram using a novel radio frequency technique with ultracold ^{6}Li gases. For all significant interaction strengths and at all temperatures we find the spin susceptibility is reduced compared to the equivalent value for a noninteracting Fermi gas. At unitarity, we can use the local density approximation to extract the integrated spin susceptibility for the uniform gas as a function of temperature, which at high temperatures is generally less than theoretically predicted. At low temperatures, our data lie within the range of theoretical predictions, although we can also describe the entire curve using a very simple one-parameter mean field model with monotonically increasing spin susceptibility.

Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms

Open physical systems with balanced loss and gain, described by non-Hermitian parity-time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {{\cal P}{\cal T}} \right)$$\end{document}PT reflection symmetric Hamiltonians, exhibit a transition which could engender modes that exponentially decay or grow with time, and thus spontaneously breaks the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry. Such \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry-breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry-breaking transitions by engineering time-periodic dissipation and coupling, which are realized through state-dependent atom loss in an optical dipole trap of ultracold ⁶Li atoms. Comparing with a single transition appearing for static dissipation, the time-periodic counterpart undergoes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry breaking and restoring transitions at vanishingly small dissipation strength in both single and multiphoton transition domains, revealing rich phase structures associated to a Floquet open system. The results enable ultracold atoms to be a versatile tool for studying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetric quantum systems.

Ultracold atoms provide controllable platforms to study many quantum mechanical phenomena. Here the authors use noninteracting fermions of ultracold Li atoms with tunable time‐periodic dissipation or coupling to demonstrate the breaking and restoration of parity‐time symmetry.

Finite-Temperature Equation of State of Polarized Fermions at Unitarity

We study in a nonperturbative fashion the thermodynamics of a unitary Fermi gas over a wide range of temperatures and spin polarizations. To this end, we use the complex Langevin method, a first principles approach for strongly coupled systems. Specifically, we show results for the density equation of state, the magnetization, and the magnetic susceptibility. At zero polarization, our results agree well with state-of-the-art results for the density equation of state and with experimental data. At finite polarization and low fugacity, our results are in excellent agreement with the third-order virial expansion. In the fully quantum mechanical regime close to the balanced limit, the critical temperature for superfluidity appears to depend only weakly on the spin polarization.

Characterizing real-space topology in Rice-Mele model by thermodynamics

The thermodynamic quantities which are related to energy-level statistics are used to characterize the real-space topology of the Rice-Mele model. Through studying the energy spectrum of the model under different boundary conditions, we found that the non-normalizable wave function for the infinite domain is reduced to the edge state adhered to the boundary. For the finite domain with symmetric boundary condition, the critical point for the topological phase transition is equal to the inverse of the domain length. In contrast, the critical point is zero for the semi-infinite domain. Additionally, the symmetry of the energy spectrum is found to be sensitive to the boundary conditions of the Rice-Mele model, and the emergence of the edge states as well as the topological phase transition can be reflected in the thermodynamic properties. A potentially practical scheme is proposed for simulating the Rice-Mele model and detecting the relevant thermodynamic quantities in the context of Bose-Einstein condensate.

Critical Velocity and Dissipation of an Ultracold Bose-Fermi Counterflow

We study the dynamics of counterflowing bosonic and fermionic lithium atoms. First, by tuning the interaction strength we measure the critical velocity v(c) of the system in the BEC-BCS crossover in the low temperature regime and we compare it to the recent prediction of Castin et al., C. R. Phys. 16, 241 (2015). Second, raising the temperature of the mixture slightly above the superfluid transitions reveals an unexpected phase locking of the oscillations of the clouds induced by dissipation.

Transition and Damping of Collective Modes in a Trapped Fermi Gas between BCS and Unitary Limits near the Phase Transition

We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion's scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.

Anomalous minimum in the shear viscosity of a Fermi gas

We measure the static shear viscosity η in a two-component Fermi gas near a broad collisional (Feshbach) resonance, as a function of interaction strength and energy. We find that η has both a quadratic and a linear dependence on the interaction strength 1/(k(FI)a), where a is the s-wave scattering length and k(FI) is the Fermi wave vector for an ideal gas at the trap center. For energies above the superfluid transition, the minimum in η as a function of interaction strength is significantly shifted toward the BEC side of resonance, to 1/(k(FI)a)≃0.25.

Imaginary polarization as a way to surmount the sign problem in ab initio calculations of spin-imbalanced Fermi gases

From ultracold atoms to quantum chromodynamics, reliable ab initio studies of strongly interacting fermions require numerical methods, typically in some form of quantum Monte Carlo calculation. Unfortunately, (non)relativistic systems at finite density (spin polarization) generally have a sign problem, such that those ab initio calculations are impractical. It is well-known, however, that in the relativistic case imaginary chemical potentials solve this problem, assuming the data can be analytically continued to the real axis. Is this feasible for nonrelativistic systems? Are the interesting features of the phase diagram accessible in this manner? By introducing complex chemical potentials, for real total particle number and imaginary polarization, the sign problem is avoided in the nonrelativistic case. To give a first answer to the above questions, we perform a mean-field study of the finite-temperature phase diagram of spin-1/2 fermions with imaginary polarization.

Observing the drop of resistance in the flow of a superfluid Fermi gas

The ability of particles to flow with very low resistance is characteristic of superfluid and superconducting states, leading to their discovery in the past century. Although measuring the particle flow in liquid helium or superconducting materials is essential to identify superfluidity or superconductivity, no analogous measurement has been performed for superfluids based on ultracold Fermi gases. Here we report direct measurements of the conduction properties of strongly interacting fermions, observing the well-known drop in resistance that is associated with the onset of superfluidity. By varying the depth of the trapping potential in a narrow channel connecting two atomic reservoirs, we observed variations of the atomic current over several orders of magnitude. We related the intrinsic conduction properties to the thermodynamic functions in a model-independent way, by making use of high-resolution in situ imaging in combination with current measurements. Our results show that, as in solid-state systems, current and resistance measurements in quantum gases provide a sensitive probe with which to explore many-body physics. Our method is closely analogous to the operation of a solid-state field-effect transistor and could be applied as a probe for optical lattices and disordered systems, paving the way for modelling complex superconducting devices.

Few-body physics with ultracold atomic and molecular systems in traps

Few-body physics has played a prominent role in atomic, molecular and nuclear physics since the early days of quantum mechanics. It is now possible-thanks to tremendous progress in cooling, trapping and manipulating ultracold samples-to experimentally study few-body phenomena in trapped atomic and molecular systems with unprecedented control. This review summarizes recent studies of few-body phenomena in trapped atomic and molecular gases, with an emphasis on small trapped systems. We start by introducing the free-space scattering properties and then investigate what happens when two particles, bosons or fermions, are placed in an external confinement. Next, various three-body systems are treated analytically in limiting cases. Our current understanding of larger two-component Fermi systems and Bose systems is reviewed, and connections with the corresponding bulk systems are established. Lastly, future prospects and challenges are discussed. Throughout this review, commonalities with other systems such as nuclei or quantum dots are highlighted.

Realization of a resonant Fermi gas with a large effective range

We have measured the interaction energy and three-body recombination rate for a two-component Fermi gas near a narrow Feshbach resonance and found both to be strongly energy dependent. Even for de Broglie wavelengths greatly exceeding the van der Waals length scale, the behavior of the interaction energy as a function of temperature cannot be described by atoms interacting via a contact potential. Rather, energy-dependent corrections beyond the scattering length approximation are required, indicating a resonance with an anomalously large effective range. For fields where the molecular state is above threshold, the rate of three-body recombination is enhanced by a sharp, two-body resonance arising from the closed-channel molecular state which can be magnetically tuned through the continuum. This narrow resonance can be used to study strongly correlated Fermi gases that simultaneously have a sizable effective range and a large scattering length.

Onset of a pseudogap regime in ultracold Fermi gases

We show, using an ab initio approach based on Quantum Monte Carlo technique, that the pseudogap regime emerges in ultracold Fermi gases close to the unitary point. We locate the onset of this regime at a value of the interaction strength corresponding to (k(F)a)(-1)≈-0.05 (a-scattering length). We determine the evolution of the gap as a function of temperature and interaction strength in the Fermi gas around the unitary limit and show that our results exhibit a remarkable agreement with the recent wave-vector resolved radio frequency spectroscopy data. Our results indicate that a finite temperature structure of the Fermi gas around unitarity is more complicated and involves the presence of the phase with preformed Cooper pairs, which, however, do not contribute to the long range order.

New field-theoretic method for the virial expansion

We develop a graphical method for computing the virial expansion coefficients for a nonrelativistic quantum field theory. As an example we compute the third virial coefficient b3 for unitary fermions, a nonperturbative system. By calculating several graphs and performing an extrapolation, we arrive at b3=-0.2930, within 0.7% of a recent computation b3=-0.290 952 95 by Liu, Hu, and Drummond, which involved summing 10,000 energy levels for three unitary fermions in a harmonic trap.

Fermi-liquid behavior of the normal phase of a strongly interacting gas of cold atoms

We measure the magnetic susceptibility of a Fermi gas with tunable interactions in the low-temperature limit and compare it to quantum Monte Carlo calculations. Experiment and theory are in excellent agreement and fully compatible with the Landau theory of Fermi liquids. We show that these measurements shed new light on the nature of the excitations of the normal phase of a strongly interacting Fermi gas.

Universal behavior of pair correlations in a strongly interacting Fermi gas

We show that short-range pair correlations in a strongly interacting Fermi gas follow a simple universal law described by Tan's relations. This is achieved through measurements of the static structure factor which displays a universal scaling proportional to the ratio of Tan's contact to the momentum C/q. Bragg spectroscopy of ultracold 6Li atoms from a periodic optical potential is used to measure the structure factor for a wide range of momenta and interaction strengths, providing broad confirmation of this universal law. We calibrate our Bragg spectra using the f-sum rule, which is found to improve the accuracy of the structure factor measurement.

Pseudogap pairing in ultracold Fermi atoms

The Bose-Einstein condensate to Bardeen-Cooper-Schrieffer crossover in ultracold Fermi gases creates an ideal environment to enrich our knowledge of many-body systems. It is relevant to a wide range of fields from condensed matter to astrophysics. The nature of pairing in strongly interacting Fermi gases can be readily studied. This aids our understanding of related problems in high-Tc superconductors, whose mechanism is still under debate due to the large interaction parameter. Here, we calculate the dynamical properties of a normal, trapped strongly correlated Fermi gas, by developing a quantum cluster expansion. Our calculations for the single-particle spectral function agree with recent rf spectroscopy measurements, and clearly demonstrate pseudogap pairing in the strongly interacting regime.

Virial expansion for a strongly correlated Fermi gas

Using a high temperature virial expansion, we present a controllable study of the thermodynamics of strongly correlated Fermi gases near the BEC-BCS crossover region. We propose a practical way to determine the expansion coefficients for both harmonically trapped and homogeneous cases, and calculate the third order coefficient b{3}(T) at finite temperatures T. At resonance, a T-independent coefficient b{3,infinity};{hom} approximately -0.290 952 95 is determined in free space. These results are compared with a recent thermodynamic measurement of 6Li atoms, at temperatures below the degeneracy temperature, and with Monte Carlo simulations.

Thermodynamics of trapped gases: generalized mechanical variables, equation of state, and heat capacity

We present the full thermodynamics of an interacting fluid confined by an arbitrary external potential. We show that for each confining potential, there emerge "generalized" volume and pressure variables V and P , that replace the usual volume and hydrostatic pressure of a uniform system. This scheme is validated with the derivation of the virial expansion of the grand potential. We discuss how this approach yields experimentally amenable procedures to find the equation of state of the fluid, P=P(VN,T) with N the number of atoms, as well as its heat capacity at constant generalized volume C_{V}=C_{V}(V,N,T) . With these two functions, all the thermodynamics properties of the system may be found. As specific examples we study weakly interacting Bose gases trapped by harmonic and by linear quadrupolar potentials within the Hartree-Fock approximation. We claim that this route provides an additional and useful tool to analyze both the thermodynamic variables of an ultracold trapped gas as well as its elementary excitations.

Critical temperature and condensate fraction of a fermion pair condensate

We report on measurements of the critical temperature and the temperature dependence of the condensate fraction for a fermion pair condensate of 6Li atoms. Bragg spectroscopy is employed to determine the critical temperature and the condensate fraction after a fast magnetic field ramp to the molecular side of the Feshbach resonance. Our measurements reveal evidence of level off of the critical temperature and limiting behavior of condensate fraction near the unitarity limit.

Observation of nearly perfect irrotational flow in normal and superfluid strongly interacting Fermi gases

We study the hydrodynamic expansion of a rotating strongly interacting Fermi gas by releasing a cigar-shaped cloud with a known angular momentum from an optical trap. As the aspect ratio of the expanding cloud approaches unity, the angular velocity increases, indicating quenching of the moment of inertia I to as low as 0.05 of the rigid body value I(rig). Remarkably, we observe this behavior in both the superfluid and collisional normal fluid regimes, which obey nearly identical zero-viscosity irrotational hydrodynamics. We attribute irrotational flow in the normal fluid to a decay of the rotational part of the stream velocity during expansion, which occurs when the shear viscosity is negligible. Using conservation of angular momentum, we directly observe a fundamental result of irrotational hydrodynamics, I/I(rig) = delta2, where delta is the deformation parameter of the cloud.

Thermodynamics of a trapped unitary Fermi gas

We present the first model-independent comparison of recent measurements of the entropy and of the critical temperature of a unitary Fermi gas, performed by Luo et al., with the most complete results currently available from finite temperature Monte Carlo calculations. The measurement of the critical temperature in a cold fermionic atomic cloud is consistent with a value T(c) = 0.23(2)epsilon(F) in the bulk, as predicted by the present authors in their Monte Carlo calculations.