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冷却フェルミ気体の輸送的性質の理論研究

啓資 藤井, Keisuke Fujii

Published Mar 1, 2021• 0 citations
OpenAlex

Abstract

Owing to their high tunability, ultracold atoms provide versatile platforms for investigating interacting systems.Here, the quantum statistics of particles is controlled by the choice of atomic species and the space geometry by the operation of optical lattices.In addition, the strength of the interparticle interaction can be tuned from weak to strong through the Feshbach resonance.One of the ultracold atomic systems which take maximum advantage of this tunability is a two-component Fermi gas near the two-body resonance, which is referred to as a resonant Fermi gas.Because the interaction is characterized by only one parameter, the s-wave scattering length, the interaction effect appears only through the scattering length near the resonance.Thus, the resonant Fermi gas shows universal behaviors independent of the details of the interaction and forms an ideal example of a strongly correlated quantum many-body system.This universality and the high tunability connect the study of ultracold atoms to various other fields of physics.In particular, a resonant Fermi gas in the unitarity limit, where the scattering length is tuned to infinity in three dimensions, shows remarkable universality because of the absence of length scales in the interaction.The emergence of the conformal invariance due to the absence of length scales is one representative example of the universality in the unitarity limit.The strong coupling, which is associated with conformality, motivates studies of the thermodynamic properties and the transport properties.For example, the system achieves an almost "perfect fluid" in the unitarity limit, where the bulk viscosity vanishes and the shear viscosity has a remarkably small ratio to the entropy density.Although the strong coupling leads to intriguing transport properties, it makes the calculations of transport coefficients difficult.This thesis investigates the transport properties of two-component Fermi gases near the two-body resonance in two and three dimensions.We particularly focus on their transport coefficients, such as the bulk viscosity, the shear viscosity, and the thermal conductivity, for an arbitrary scattering length.In order to carry out reliable analysis in the strongly correlated regime, such as near the unitarity limit, we need non-perturbative methods.One of the nonperturbative methods is to use the conformal symmetry that emerges in the unitarity limit.The application of this symmetry-based approach to hydrodynamics revealed the vanishing bulk viscosity in the unitarity limit.Another non-perturbative approach is an expansion in terms of the fugacity, called the quantum virial expansion.In the high-temperature and low-density regime, this expansion is valid because the fugacity is small.First of all, we focus on the bulk viscosity, which characterizes the dissipation caused by an expansion of the fluid volume.The vanishing bulk viscosity in the unitarity limit is intuitively understood because the entropy does not change before and after an isotropic expansion of the i List of Publications Publications relevant to this thesis 1.

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Owing to their high tunability, ultracold atoms provide versatile platforms for investigating interacting systems.Here, the quantum statistics of particles is controlled by the choice of atomic species and the space geometry by the operation of optical lattices.In addition, the strength of the interparticle interaction can be tuned from weak to strong through the Feshbach resonance.One of the ultracold atomic systems which take maximum advantage of this tunability is a two-component Fermi gas near the two-body resonance, which is referred to as a resonant Fermi gas.Because the interaction is characterized by only one parameter, the s-wave scattering length, the interaction effect appears only through the scattering length near the resonance.Thus, the resonant Fermi gas shows universal behaviors independent of the details of the interaction and forms an ideal example of a strongly correlated quantum many-body system.This universality and the high tunability connect the study of ultracold atoms to various other fields of physics.In particular, a resonant Fermi gas in the unitarity limit, where the scattering length is tuned to infinity in three dimensions, shows remarkable universality because of the absence of length scales in the interaction.The emergence of the conformal invariance due to the absence of length scales is one representative example of the universality in the unitarity limit.The strong coupling, which is associated with conformality, motivates studies of the thermodynamic properties and the transport properties.For example, the system achieves an almost "perfect fluid" in the unitarity limit, where the bulk viscosity vanishes and the shear viscosity has a remarkably small ratio to the entropy density.Although the strong coupling leads to intriguing transport properties, it makes the calculations of transport coefficients difficult.This thesis investigates the transport properties of two-component Fermi gases near the two-body resonance in two and three dimensions.We particularly focus on their transport coefficients, such as the bulk viscosity, the shear viscosity, and the thermal conductivity, for an arbitrary scattering length.In order to carry out reliable analysis in the strongly correlated regime, such as near the unitarity limit, we need non-perturbative methods.One of the nonperturbative methods is to use the conformal symmetry that emerges in the unitarity limit.The application of this symmetry-based approach to hydrodynamics revealed the vanishing bulk viscosity in the unitarity limit.Another non-perturbative approach is an expansion in terms of the fugacity, called the quantum virial expansion.In the high-temperature and low-density regime, this expansion is valid because the fugacity is small.First of all, we focus on the bulk viscosity, which characterizes the dissipation caused by an expansion of the fluid volume.The vanishing bulk viscosity in the unitarity limit is intuitively understood because the entropy does not change before and after an isotropic expansion of the i List of Publications Publications relevant to this thesis 1.
We measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave collisional (Feshbach) resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and transport coefficients are universal functions of density n and temperature T. We present a new calibration of the temperature as a function of global energy, which is directly measured from the cloud profiles. Using the calibration, the trap-averaged shear viscosity in units of ℏn is determined as a function of the reduced temperature at the trap center, from nearly the ground state to the unitary two-body regime. Low-temperature data are obtained from the damping rate of the radial breathing mode, whereas high-temperature data are obtained from hydrodynamic expansion measurements. We also show that the best fit to the high-temperature expansion data is obtained for a vanishing bulk viscosity. The measured trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of shear viscosity to entropy density, which is compared with that conjectured for a perfect fluid.
We measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave collisional (Feshbach) resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and transport coefficients are universal functions of density n and temperature T . We present a new calibration of the temperature as a function of global energy, which is directly measured from the cloud profiles. Using the calibration, the trap-averaged shear viscosity in units of ℏ n is determined as a function of the reduced temperature at the trap center, from nearly the ground state to the unitary two-body regime. Low-temperature data are obtained from the damping rate of the radial breathing mode, whereas high-temperature data are obtained from hydrodynamic expansion measurements. We also show that the best fit to the high-temperature expansion data is obtained for a vanishing bulk viscosity. The measured trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of shear viscosity to entropy density, which is compared with that conjectured for a perfect fluid.
This dissertation describes the first experimental measurement of the energy and interaction dependent shear viscosity eta\\etaeta and bulk viscosity zeta\\zetazeta in the hydrodynamic expansion of a two-component Fermi gas near a broad collisional (Feshbach) resonance. This expansion also provides a precise test of scale invariance and an examination of local thermal equilibrium as a function of interaction strength. After release from an anisotropic optical trap, we observe that a resonantly interacting gas obeys scale-invariant hydrodynamics, where the mean square cloud size langlemathbfr2rangle=langlex2+y2+z2rangle\\langle{\\mathbf{r}}^2\\rangle=\\langle x^2+y^2+z^2\\ranglelanglemathbfr2rangle=langlex2+y2+z2rangle expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero, 0.00(0.04),hbar,n0.00(0.04)\\,\\hbar\\,n0.00(0.04),hbar,n, with nnn the density. In contrast, the aspect ratios of the cloud exhibit anisotropic ``elliptic" flow with an energy-dependent shear viscosity. Tuning away from resonance, we observe conformal symmetry breaking, where langlemathbfr2rangle\\langle{\\mathbf{r}}^2\\ranglelanglemathbfr2rangle deviates from ballistic flow. We find that eta\\etaeta has both a quadratic and a linear dependence on the interaction strength 1/(kFIa)1/({k_{FI}a})1/(kFI​a), where aaa is the s-wave scattering length and kFIk_{FI}kFI​ is the Fermi wave vector for an ideal gas at the trap center. At low energy, the minimum is less than the resonant value and is significantly shifted toward the BEC side of resonance, to 1/(kFIa)=0.21/(k_{FI}a) = 0.21/(kFI​a)=0.2.
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary ppp-wave Fermi gas.
Feshbach resonances of trapped ultracold alkali-metal atoms allow to vary the atomic scattering length aaa. At very large values of aaa the system enters an universal strongly coupled regime in which its properties---the ground-state energy, pressure, etc.---become independent of aaa. We discuss the transport properties of such systems. In particular, the universality arguments imply that the shear viscosity of ultracold Fermi atoms at the Feschbach resonance is proportional to the particle number density nnn and the Plank constant $\ensuremath{\hbar}$: $\ensuremath{\eta}=\ensuremath{\hbar}n{\ensuremath{\alpha}}_{\ensuremath{\eta}}$, where ${\ensuremath{\alpha}}_{\ensuremath{\eta}}$ is a universal constant. Using Heisenberg uncertainty principle and Einstein's relation between diffusion and viscosity we argue that the viscosity has the lower bound given by ${\ensuremath{\alpha}}_{\ensuremath{\eta}}\ensuremath{\leqslant}{(6\ensuremath{\pi})}^{\ensuremath{-}1}$. We relate the damping of low-frequency density oscillations of ultracold optically trapped 6Li^{6}\mathrm{Li}6Li atoms to viscosity and find that the value of the coefficient ${\ensuremath{\alpha}}_{\ensuremath{\eta}}$ is about 0.3. We also show that such a small viscosity cannot be explained by kinetic theory based on binary scattering. We conclude that the system of ultracold atoms near the Feshbach resonance is a near-ideal liquid.
We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, nnn is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.