We study strongly interacting two component Fermi gas near a Feshbach resonance. By using a ground state energy functional constructed based on asymptotic limits and Monte Carlo calculations, we calculate the contact, structure factor, and collective oscillation frequencies in the BCS-BEC crossover region. The calculated contact and structure factor show excellent agreements with recent experiments. We compare these results with a standard mean-field theory and find that the contact is proportional to the square of superfluid order parameter. Further, we present the chemical potential and the polytropic index in terms of homogenous energy and the contact.
The Anderson-Bogoliubov branch of collective excitations in a condensed Fermi gas is treated using the effective bosonic action of Gaussian pair fluctuations. The spectra of collective excitations are treated for finite temperature and momentum throughout the BCS-BEC crossover. The obtained spectra explain, both qualitatively and quantitatively, recent experimental results on Goldstone modes in atomic Fermi superfluids.
We theoretically investigate the collective modes of imbalanced two component one-dimensional Fermi gases with attractive interactions. This is done for trapped and untrapped systems both at zero and non-zero temperature, using self-consistent mean-field theory and the random phase approximation. We discuss how the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state can be detected and the periodicity of the associated density modulations determined from its collective mode spectrum. We also investigate the accuracy of the single mode approximation for low-lying collective excitations in a trap by c…
We investigate the collective modes in the spin-orbit coupled Fermi gas with repulsive s-wave interaction. The interplay between spin-orbit coupling and atom-atom interactions plays the crucial role in the collective behaviors of Fermi gas. In contrast with ordinary Fermi liquid, spin-orbit coupled Fermi gas has strongly correlated spin and density excitations. Within the scheme of random phase approximation, we classify collective modes based on the symmetry group and determine their properties via the poles of corresponding correlation functions. Besides, the particle-hole continuum is obtai…
Topological energy bands have important geometrical properties described by the Berry curvature. We show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field. Using an operator approach, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Colle…
We show that the wave function of a one dimensional spinor gas with contact $s$-wave interaction, either bosonic or fermionic, can be mapped to the direct product of the wave function of a spinless Fermi gas with short-range $p$-wave interaction and that of a spin system governed by spin parity projection operators. Applying this mapping to strongly interacting spinor gases, we obtain a generalized spin chain model that captures both the static and dynamics properties of the system. Using this spin chain model, we investigate the breathing mode frequency and the quench dynamics of strongly int…
We analyze the macroscopic dynamics of a Bose gas axially confined in an optical lattice with a superimposed harmonic trap, taking into account weak tunneling effect. Our results show that upon transition to the quasi-two-dimensional (2D) regime of the trapped gas, the 3D equation of state and equilibrium density profile acquire corrections from 2D many-body effects. The corresponding frequency shift in the transverse breathing mode is accessible within current facilities, suggesting a direct observation of dimensional effects. Comparisons with other relevant effects are also presented.
We suggest that collective oscillation frequencies of cold trapped gases can be used to test predictions from quantum many-body physics. Our motivation lies both in rigid experimental tests of theoretical calculations and a possible improvement of measurements of particle number, chemical potential or temperature. We calculate the effects of interaction, dimensionality and thermal fluctuations on the collective modes of a dilute Bose gas in the hydrodynamic limit. The underlying equation of state is provided by non-perturbative Functional Renormalization Group or by Lee--Yang theory. The spect…
Using both fluid and kinetic descriptions, where repulsive forces between near by atoms are included, we discuss the basic oscillations and waves of a cloud of ultra-cold atoms confined in a magneto-optical trap. The existence of a hybrid mode, with properties similar to both plasma and acoustic waves is described in detail. Tonks-Dattner resonances for confined hybrid modes in a spherical cloud are discussed and the prediction of a nonlinear coupling between the dipole resonanc and the hybrid modes is considered. Landau damping processes and quasi-linear diffusion in velocity space are also d…
Recently, a seminal STIRAP experiment allowed the creation of 40K-87Rb molecules in the rovibrational ground state [K.-K. Ni et al., Science 322, 231 (2008)]. In order to describe such a polarized dipolar Fermi gas in the hydrodynamic regime, we work out a variational time-dependent Hartree-Fock approach. With this we calculate dynamical properties of such a system as, for instance, the frequencies of the low-lying excitations and the time-of-flight expansion. We find that the dipole-dipole interaction induces anisotropic breathing oscillations in momentum space. In addition, after release fro…
We study the transition from hydrodynamic to collisionless behavior in collective modes of ultracold trapped Fermi gases. To that end, we solve the Boltzmann equation for the trapped Fermi gas via the moments method. We showed previously that it is necessary to go beyond second-order moments if one wants to reproduce the results of a numerical solution of the Boltzmann equation. Here, we will give the detailed description of the method including fourth-order moments. We apply this method to the case of realistic parameters, and compare the results for the radial quadrupole and scissors modes a…
We study the phononic collective modes of the pairing field $Δ$ and their corresponding signature in both the order-parameter and density response functions for a superfluid Fermi gas at all temperatures below $T_c$ in the collisionless regime. The spectra of collective modes are calculated within the Gaussian Pair Fluctuation approximation. We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a non-perturbative analytic continuation of the pairing field propagator. At low temperature, we recover the known exponential temperat…
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature $T_c$ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below $T_c$ in the system's response to a driving pairing field. When approaching $T_c$ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a very low energy-momentum region when the pair correlation length reaches its critical divergent behavior $ξ_{\rm pair}\propto|T_c-T|^{-1/2}$; elsewhere, they are repla…
We theoretically analyze interference patterns of parametrically driven one dimensional ultracold atomic gases. By modulating the interaction strength periodically in time, we propose to excite collective modes in a pair of independent one dimensional gases at energies corresponding to the drive frequency. The excited collective modes lead to spatial oscillations in the correlations of the interference pattern, which can be analyzed to obtain the sound velocity of the collective modes. We discuss both bosonic and fermionic systems, and how such experiments could be used to probe spin charge se…
We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors. At zero temperature, we analytically continue the equation on the collective mode energy in Anderson's Random Phase Approximation or Gaussian fluctuations through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. The branch exists as long as the chemical potential $μ$ is positive and the wave number below $\sqrt{2mμ}/\hbar$ (with m the fermion mass). In the long wave…
Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high temperature regimes. A cluster expansion that describes the properties of the N-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the hi…
Very different strongly interacting quantum systems such as Fermi gases, quark-gluon plasmas formed in high energy ion collisions and black holes studied theoretically in string theory are known to exhibit quantitatively similar damping of hydrodynamic modes. It is not known if such similarities extend beyond the hydrodynamic limit. Do non-hydrodynamic collective modes in Fermi gases with strong interactions also match those from string theory calculations? In order to answer this question, we use calculations based on string theory to make predictions for novel types of modes outside the hydr…
We calculate the collective modes of ultracold trapped alkaline-earth fermionic atoms, which possess an SU($N$) symmetry of the nuclear spin degree of freedom, and a controllable $N$, with $N$ as large as $10$. We calculate the breathing and quadrupole modes of two-dimensional and three-dimensional harmonically trapped gases in the normal phase. We particularly concentrate on two-dimensional gases, where the shift is more accessible experimentally, and the physics has special features. We present results as a function of temperature, interaction strength, density, and $N$. We include calculati…
The ground states and collective excitations of trapped Fermion gases consisting of atoms with magnetic dipole moment are studied using a time-dependent density-matrix approach. The advantages of the density-matrix approach are that one-body and two-body observables are directly calculated using one-body and two-body density matrices and that it has a clear relation to the Hartree-Fock (HF) and time-dependent HF theory. The HF calculations show the magnetization of the gases when the dipole-dipole interaction is strong. It is shown that the tensor properties of the dipole-dipole interaction ar…
Collective phenomena in quantum many-body systems are often described in terms of hydrodynamics, an appropriate framework when the involved particle numbers are effectively macroscopic. We propose to use experiments on expanding clouds of few and many interacting cold atoms to investigate the emergence of hydrodynamics as a function of particle number. We consider gases confined in two-dimensional elliptically-deformed traps, and we employ the manifestation of elliptic flow as an indicator of collective behavior. We quantify the response of the gas to the deformation of the trapping potential,…
The collective dynamics of a dipolar fermionic quantum gas confined in a one-dimensional double-well superlattice is explored. The fermionic gas resides in a paramagnetic-like ground state in the weak interaction regime, upon which a new type of collective dynamics is found when applying a local perturbation. This dynamics is composed of the local tunneling of fermions in separate supercells, and is a pure quantum effect, with no classical counterpart. Due to the presence of the dipolar interactions the local tunneling transports through the entire superlattice, giving rise to a collective dyn…
In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as $\sim 1/|p|^4$ at large momentum $p$. Tan's relation connects the amplitude of that $1/|p|^4$ tail to the adiabatic derivative of the energy with respect to the gas' coupling constant or scattering length. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution whic…
We study mass and population imbalance effect on density (in-phase) and spin (out-of-phase) collective modes in a two-component normal Fermi gas. By calculating eigenmodes of the linearized Boltzmann equation as well as the density/spin dynamic structure factor, we show that mass and population imbalance effects offer a variety of collective mode crossover behaviors from collisionless to hydrodynamic regimes. The mass imbalance effect shifts the crossover regime to the higher-temperature, and a significant peak of the spin dynamic structure factor emerges only in the collisionless regime. This…
We numerically solve the Boltzmann equation for trapped fermions in the normal phase using the test-particle method. After discussing a couple of tests in order to estimate the reliability of the method, we apply it to the description of collective modes in a spherical harmonic trap. The numerical results are compared with those obtained previously by taking moments of the Boltzmann equation. We find that the general shape of the response function is very similar in both methods, but the relaxation time obtained from the simulation is significantly longer than that predicted by the method of m…
We investigate universal properties of one-dimensional multi-component systems comprised of fermions, bosons, or an arbitrary mixture, with contact interactions and subjected to an external potential. The masses and the coupling strengths between different types of particles are allowed to be different and we also take into account the presence of an arbitrary magnetic field. We show that the momentum distribution of these systems exhibits a universal $n_σ(k) \sim C_σ/k^4$ decay with $C_σ$ the contact of species $σ$ which can be computed from the derivatives of an appropriate thermodynamic pot…
Path-integral analyses originally pioneered in the study of the complex-phase problem afflicting lattice calculations of finite-density quantum chromodynamics are generalized to non-relativistic Fermi gases with repulsive interactions. Using arguments similar to those previously applied to relativistic theories, we show that the analogous problem in nonrelativistic systems manifests itself naturally in Tan's contact as a nontrivial cancellation between terms with varied dependence on extensive thermodynamic quantities. We analyze that case under the assumption of gaussian phase distribution, w…
We study collective modes of superfluid Bose gases in optical lattices at commensurate fillings. We focus on the vicinity of the quantum phase transition to the Mott insulator, where there exists the Higgs amplitude mode in addition to the Nambu-Goldstone phase mode associated with the spontaneous U(1) symmetry breaking. We analyze finite-temperature effects on the damping of the collective modes by using an effective spin-1 model and the field theoretical methods based on the finite-temperature Green's function. We calculate the damping rates up to 1-loop order and evaluate them analytically …
We investigate collective excitations of density fluctuations and a dynamic density structure factor in a mixture of Bose and Fermi gases in a normal phase. With decreasing temperature, we find that the frequency of the collective excitation deviates from that of the hydrodynamic sound mode. Even at temperature much lower than the Fermi temperature, the collective mode frequency does not reach the collisionless limit analogous to zero sound in a Fermi gas, because of collisions between bosons and fermions.
The low-energy modes of a superfluid atomic Fermi gas at zero temperature are investigated. The Bose-Einstein-condensate (BEC) side of the superfluid phase is studied in detail. The atoms are assumed to be in only one internal state, so that for a sufficiently diluted gas the pairing of fermions can be considered effective in the l=1 channel only. In agreement with previous works on p-wave superfluidity in Fermi systems, it is found that the $p_x+ip_y$ phase represents the lowest energy state in both the Bardeen-Cooper-Schrieffer (BCS) and BEC sides. Our calculations show that at low energy th…
Motivated by recent experimental advances in creating polar molecular gases in the laboratory, we theoretically investigate the many body effects of two-dimensional dipolar systems with the anisotropic and $1/r^3$ dipole-dipole interactions. We calculate collective modes of 2D dipolar systems, and also consider spatially separated bilayer and multilayer superlattice dipolar systems. We obtain the characteristic features of collective modes in quantum dipolar gases. We quantitatively compare the modes of these dipolar systems with the modes of the extensively studied usual two-dimensional elect…
We measure collective excitations of a harmonically trapped two-dimensional (2D) SU($N$) Fermi gas of $^{173}$Yb confined to a stack of layers formed by a one-dimensional optical lattice. Quadrupole and breathing modes are excited and monitored in the collisionless regime $\lvert\ln(k_F a_{2D})\rvert\gg 1$ with tunable spin. We observe that the quadrupole mode frequency decreases with increasing number of spin components due to the amplification of the interaction effect by $N$ in agreement with a theoretical prediction based on 2D kinetic equations. The breathing mode frequency, however, is m…
Itinerant ferromagnetism in cold Fermi gases with repulsive interactions is studied applying the Jastrow-Slater approximation generalized to finite polarization and temperature. For two components at zero temperature a second order transition is found at $ak_F\simeq0.90$ compatible with QMC. Thermodynamic functions and observables such as the compressibility and spin susceptibility and the resulting fluctuations in number and spin are calculated. For trapped gases the resulting cloud radii and kinetic energies are calculated and compared to recent experiments. Spin polarized systems are recomm…
We present a theoretical study of the superfluidity and the corresponding collective modes in two-component atomic Fermi gases with s-wave attraction and synthetic Rashba spin-orbit coupling. The general effective action for the collective modes is derived from the functional path integral formalism. By tuning the spin-orbit coupling from weak to strong, the system undergoes a crossover from an ordinary BCS/BEC superfluid to a Bose-Einstein condensate of rashbons. We show that the properties of the superfluid density and the Anderson-Bogoliubov mode manifest this crossover. At large spin-orbit…
A characteristic property of superfluidity and -conductivity is the presence of quantized vortices in rotating systems. To study the BEC-BCS crossover the two most common methods are the Bogoliubov-De Gennes theory and the usage of an effective field theory. In order to simplify the calculations for one vortex, it is often assumed that the hyperbolic tangent yields a good approximation for the vortex structure. The combination of a variational vortex structure, together with cylindrical symmetry yields analytic (or numerically simple) expressions. The focus of this article is to investigate to…
We investigate the fermionic quasiparticle branch of superfluid Fermi gases in the BCS-BEC crossover and calculate the quasiparticle lifetime and energy shift due to its coupling with the collective mode. The only close-to-resonance process that low-energy quasiparticles can undergo at zero temperature is the emission of a bosonic excitation from the phononic branch. Close to the minimum of the branch we find that the quasiparticles remain undamped, allowing us to compute corrections to experimentally relevant quantities such as the energy gap, location of the minimum, effective mass, and Land…
We study the quench dynamics of a two-component ultracold Fermi gas from the weak into the strong interaction regime, where the short time dynamics are governed by the exponential growth rate of unstable collective modes. We obtain an effective interaction that takes into account both Pauli blocking and the energy dependence of the scattering amplitude near a Feshbach resonance. Using this interaction we analyze the competing instabilities towards Stoner ferromagnetism and pairing.
It is generally believed that the BCS-BEC evolution in fermionic systems with s-wave pairing is a smooth crossover. However, for nonzero orbital-angular-momentum pairing such as p- or d-wave pairing, the system undergoes a quantum phase transition at the point where the chemical potential $μ$ vanishes. In this paper, we study the BCS-BEC quantum phase transition and the collective excitations associated with the order-parameter fluctuations in two-dimensional fermionic systems with p- and d-wave pairings. We show that the quantum phase transition in such systems can be generically traced back …
Advanced data analysis techniques have proved to be crucial for extracting information from noisy images. Here we show that principal component analysis can be successfully applied to ultracold gases to unveil their collective excitations. By analyzing the correlations in a series of images we are able to identify the collective modes which are excited, determine their population, image their eigenfunction, and measure their frequency. Our method allows to discriminate the relevant modes from other noise components and is robust with respect to the data sampling procedure. It can be extended t…
We develop a method to extract the dissipation for a heavy moving impurity immersed in superfluid Fermi gases. The drag force is derived analytically. As a reward, we are able to extract the dynamical structure factor, from which density excitations of the system is carefully examined. We show that dissipations through drag force is associated with two types of excitations, one being single-particle and the other being collective. We map out the critical velocity for dissipation across the BEC-BCS crossover, consistent with existing experiments. For a magnetic impurity, we show that the dissip…
The properties of two-component Fermi gases with zero-range interactions are universal. We use an explicitly correlated Gaussian basis set expansion approach to investigate small equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement. At unitarity, we determine the ground state energy for systems with up to ten particles interacting through finite-range two-body potentials for both even and odd number of particles. We extrapolate the energies to the zero-range limit using a novel scheme that removes the linear and, in some cases, also the quadratic depen…
The contact parameter in unitary Fermi Gases governs the short-distance, high-momentum, and high-energy properties of the system. We perform accurate quantum Monte Carlo calculations with highly optimized trial functions to precisely determine this parameter at T=0, demonstrate its universal application to a variety of observables, and determine the regions of momentum and energy over which the leading short-range behavior is dominant. We derive Tan's expressions for the contact parameter using just the short-range behavior of the ground-state many-body wave function, and use this behavior to …
The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In p…
Collective excitations of driven-dissipative quantum fluids of light
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；ENS-PSL Research University（巴黎高等师范学院-巴黎文理研究大学）；Collège de France（法兰西公学院）；INO-CNR（意大利国家研究委员会光学研究所）；Università di Trento（特伦托大学）
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particul…
Tunability of effective two body interactions near Feshbach resonances is a powerful experimental tool in systems of ultracold atoms. It has been used to explore a variety of intriguing phenomena in recent experiments. However not all of the many-body properties of such systems can be understood in terms of effective models with contact interaction given by the scattering length of the two particles in vacuum. For example, when a two component Fermi mixture is quenched to the BEC side of the Feshbach resonance, a positive scattering length suggests that interactions are repulsive and thus coll…
We study Tan's contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan's contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expr…
We study the collective modes of the minority component of a highly unbalanced Bose-Bose mixtures. In the miscible case the minority component feels an effective external potential and we derive an analytical expression for the mode frequencies. The latter is independent of the minority component interaction strength. In the immiscible case we find that the ground state can be a two-domain walls soliton. Although the mode frequencies are continuous at the transition, their behaviour is very different with respect to the miscible case. The dynamical behaviour of the solitonic structure and the …
Beyond spin-charge separation: Helical modes and topological quantum phase transitions in one-dimensional Fermi gases with spin-orbit and Rabi couplings
Weakly interacting Fermi gases exhibit rich collective dynamics in spin-dependent potentials, arising from correlations between spin degrees of freedom and conserved single atom energies, offering broad prospects for simulating many-body quantum systems by engineering energy-space "lattices," with controlled energy landscapes and site to site interactions. Using quantum degenerate clouds of $^6$Li, confined in a spin-dependent harmonic potential, we measure complex, time-dependent spin-density profiles, varying on length scales much smaller than the cloud size. We show that a one-dimensional m…
The Berezinskii-Kosterlitz-Thouless transition in 2D dipolar systems has been studied recently by path integral Monte Carlo (PIMC) simulations [A. Filinov et al., PRL 105, 070401 (2010)]. Here, we complement this analysis and study temperature-coupling strength dependence of the density (particle-hole) and single-particle (SP) excitation spectra both in superfluid and normal phases. The dynamic structure factor, S(q,omega), of the longitudinal excitations is rigorously reconstructed with full information on damping. The SP spectral function, A(q,omega), is worked out from the one-particle Mats…
We show that arbitrarily high-order exceptional points (EPs) can be achieved in a repulsively interacting two-species Bose gas in one dimension. By exactly solving the non-Hermitian two-boson problem, we demonstrate the existence of third-order EPs when the system is driven across the parity-time symmetry breaking transition. We further address the fourth-order EPs with three bosons and generalize the results to $N$-body system, where the EP order can be as high as $N+1$. Physically, such high order originates from the intrinsic ferromagnetic correlation in spinor bosons, which renders the ent…
The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases.We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature.Combining thermal Bethe ansatz, local-density approximation, and exact quantum Monte Carlo calculations,we show that the contact is a universal function of only two scaling parameters, and determine the scaling f…
We study the rotational properties of a two-component Bose-Einstein condensed gas of distinguishable atoms which are confined in a ring potential using both the mean-field approximation, as well as the method of diagonalization of the many-body Hamiltonian. We demonstrate that the angular momentum may be given to the system either via single-particle, or "collective" excitation. Furthermore, despite the complexity of this problem, under rather typical conditions the dispersion relation takes a remarkably simple and regular form. Finally, we argue that under certain conditions the dispersion re…
