Measuring the Hydrodynamic Linear Response of a Unitary Fermi Gas
Lorin Baird Xin Wang Stetson Roof J. E. Thomas






^1Department of  Physics, North Carolina State University, Raleigh, NC 27695, USA




We directly observe the hydrodynamic linear  response of a unitary Fermi gas confined in a box potential and subject to a spatially periodic optical potential that is translated into the cloud at speeds ranging from subsonic to supersonic. We show that the time-dependent change of the density profile is sensitive  to the thermal conductivity, which controls the relaxation rate of the temperature gradients and hence the responses arising from adiabatic and isothermal compression.




Measuring the Hydrodynamic Linear Response of a Unitary Fermi Gas
    Lorin Baird, Xin Wang, Stetson Roof and J. E. Thomas
    December 30, 2023
=================================================================

A unitary Fermi gas is a scale-invariant, strongly interacting quantum many-body system, created by tuning a trapped, two-component cloud near a collisional (Feshbach) resonance <cit.>. Unitary gases are of great interest <cit.>, as the thermodynamic properties and transport coefficients are universal functions of the density and temperature, enabling parameter-free comparisons with predictions. Equilibrium thermodynamic properties of trapped unitary gases have been well characterized <cit.>. In contrast, hydrodynamic transport measurements require dynamical experiments that have been obscured by the low density near the cloud edges, which leads to free streaming. For expanding clouds <cit.>, this problem has been circumvented by employing second order hydrodynamics methods to extract the local shear viscosity <cit.>, and is obviated for trapped samples with uniform density. A normal unitary gas, at temperatures above the superfluid transition, is a single component fluid that affords the simplest universal system for hydrodynamic transport measurements, as the transport properties comprise only the shear viscosity η and the thermal conductivity κ_T, since the bulk viscosity vanishes in scale-invariant systems <cit.>. Further, measurements in the normal fluid at high temperature T can be compared with benchmark variational calculations for a unitary gas in the two-body Boltzmann limit <cit.>,

    η=15/32√(π)(mk_BT)^3/2/ħ^2

and

    κ_T=15/4k_B/m η,

with k_B the Boltzmann constant and m the atom mass.

In this Letter, we demonstrate a new probe of hydrodynamic transport, which is applied to a normal unitary Fermi gas of ^6Li. The gas is confined in a repulsive box potential, creating a sample of nearly uniform density, and driven by a moving, spatially periodic optical potential of chosen wavelength λ along one axis z, which moves into the box at a selected speed v. We measure the density response δ n(z,t), which is analyzed using a linear hydrodynamics model. The model shows that the response profiles are sensitive to the effective sound speed, which is controlled by the ratio of the tunable wave frequency ω=2π v/λ to the decay rate of the temperature gradients, γ_κ∝κ_T/λ^2.  When γ_κ<<ω, temperature gradients relax slowly and sound waves propagate at the adiabatic sound speed c_0. In the opposite limit, γ_κ>>ω, temperature gradients relax quickly and sound waves propagate at the isothermal sound speed c_T<c_0.



The experiments, Fig. <ref>, employ ultracold ^6Li atoms, in a balanced mixture of the two lowest hyperfine states, which are loaded into a box potential U_0, comprising six sheets of blue-detuned light, created by two digital micromirror devices (DMDs). This produces a rectangular density profile with dimensions (129×84×58) μm, which slowly varies due to the curvature of the bias magnetic field,  Fig. <ref>. The average total central density is n_0≃ 2.6×10^11 atoms/cm^3, for which the Fermi energy ϵ_F0≡ k_B T_F =k_B× 0.16 μK and the Fermi speed v_F=2.1 cm/s. As suggested by Zhang and Yu <cit.>, we probe the linear response δ n(z,t) by employing one of the DMDs to generate a small spatially periodic optical potential that moves through the cloud at speed v,

    δ U(z,t)=δ U_0[1-ϵcos(qz-qvt)] H(vt-z),

where q=2π/λ. The Heaviside envelope function H(vt-z) vanishes inside the box at t=0. Positive light intensity requires 1-ϵcos(qz-qvt)≥ 0, so that ϵ≤ 1.  For each speed v, δ U(z,t) is turned on for a fixed number of periods, after which an absorption image is recorded to obtain the column density. For the longest wavelength employed in the experiments, λ = 30 μm, the image is taken after three periods (leading edge at 90 μm), while for the shortest wavelength λ = 19 μm, imaging occurs after four periods (leading edge at 76 μm). Instead of measuring the energy input, as proposed in ref. <cit.>, we directly obtain the response δ n(z,t)/n_0 from the integrated column density, which is measured five times for each λ at several different frequencies f≡ v/λ from 200 to 800 Hz.

Figs. <ref>, <ref>, and Fig. <ref> show the density response δ n(z,t) as the drive speed v is varied from subsonic v<c_0 to supersonic v>c_0, where c_0 is the adiabatic sound speed. At low drive speeds, the leading edge of the response is nearly flat, as sound waves propagate well past the front of the driving potential. As v approaches c_0, the amplitude of the density response increases and the leading peak narrows.



To understand the density profiles arising from the perturbation δ U, we construct the coupled equations for the change in the density δ n and for the change in the entropy  per particle δ s_1. The analysis is simplified for experiments in the linear response regime, where <cit.>

    ∂_t^2δ n-c_0^2 (∂_z^2δ n+∂_z^2δs̃_1)
    -4/3 η/n_0 m ∂_z^2∂_tδ n   =   
    1/m ∂_z[n_0(z) ∂_zδ U+δ n ∂_z U_0(z)],

with  m the atom mass. Here, δs̃_1=n_0β T_0 δ s_1/c_P, with β the thermal expansivity and T_0 the initial sample temperature <cit.>. We find

    ∂_tδs̃_1-κ_T/n_0 c_V ∂_z^2δs̃_1=
    κ_T/n_0 c_Vc_P-c_V/c_P ∂_z^2δ n,

where c_V and c_P are the heat capacities per particle at constant volume and at constant pressure, determined from the measured equation of state <cit.>. On the left side of eq. <ref>, the c_0^2 terms arise from the pressure change δ p <cit.>. The η term produces a viscous damping rate γ_η=4η  q^2/(3n_0 m) for the response of the density to the spatially periodic part of δ U(z,t), eq. <ref>. On the right hand side of eq. <ref>, the first term  arises from the perturbing potential, with n_0(z) the  background density, which varies slowly due to the bias magnetic field curvature. Here, we retain the full spatial variation of the force per unit volume <cit.>, which vanishes at the box edges. In the second term, ∂_zU_0(z) is the force from the box potential. We determine ∂_z U_0(z) from n_0(z), which is measured in equilibrium <cit.>.


In addition to the shear viscosity, δ n(z,t) carries information about the thermal conductivity κ_T, which sets the relaxation rate, γ_κ=κ_T q^2/(n_0c_V)  in eq. <ref>, of the spatially periodic temperature profile that is imprinted by δ U(z,t),  eq. <ref>. For a high speed v, the wave frequency q v>>γ_κ. Then ∂_tδs̃_1 dominates in eq. <ref> and δs̃_1≃ 0, yielding (∂_t^2-c_0^2 ∂_z^2) δ n on the left side of eq. <ref>. In this case, the compression is adiabatic, and sound waves propagate at the speed c_0. In the opposite limit of a low speed v, the wave frequency qv<<γ_κ. Eq. <ref> shows that ∂_z^2δs̃_1≃-(c_P-c_V)/c_P ∂_z^2 δ n, yielding  (∂_t^2-c_T^2 ∂_z^2) δ n  in eq. <ref>, with c_T=c_0√(c_V/c_P). Then, the compression is isothermal and sound waves propagate at the isothermal sound speed c_T <cit.>.

To model the normal fluid data,  c_0 is used as a fit parameter. The fitted c_0 then serves as a thermometer, as the reduced temperature θ_0 = T_0/T_F of the gas is monotonically related to c_0/v_F in the normal fluid regime <cit.>. With θ_0 determined, c_V and c_P are then fixed by the measured equation of state <cit.>. Further, θ_0 determines the shear viscosity η as discussed below.


Our analysis benefits from recent progress in determining the local shear viscosity of the normal fluid from hydrodynamic expansion experiments <cit.>. Extraction of η is simplified in expansion measurements, because the temperature gradient is negligible <cit.> so that the thermal conductivity κ_T can be neglected.  The most complete data for the shear viscosity have been obtained from the aspect ratio of expanding cigar-shaped clouds, measured at a fixed time t after release from an optical trap as a function of the cloud energy <cit.>. The latest hydrodynamic analysis utilizes an anisotropic pressure model, which properly interpolates between the hydrodynamic behavior in the dense regions of the cloud and the free streaming ballistic expansion near the cloud edges <cit.>.  The new analysis yields an expansion of the local shear viscosity in powers of the diluteness nλ_T^3,

    η=η_0(mk_BT)^3/2/ħ^2 [1+η_2(nλ_T^3)+...],

where λ_T=h/√(2π mk_BT) is the thermal wavelength and n is the total density for a balanced two-component mixture. Fits to the expansion data yield η_0=0.265(20), in excellent agreement with the variational result obtained from the two-body Boltzmann equation for a unitary gas, eq. <ref>, η_0=15/(32√(π))=0.26446 <cit.>. This confirms that the data and the analysis properly reproduce the high temperature limit, which is independent of the density. The next order term is independent of the temperature, with η_2=0.060(20), while the η_3(nλ_T^3)^2 term is  negligible. Remarkably, the first two terms fit the expansion data down to temperatures just above the superfluid transition. We therefore use eq. <ref> as in input for eq. <ref>, where η/(n_0 m)≡α(θ_0) ħ/m and α(θ_0)=α_0 θ_0^3/2+α_2, with α_0=(3π^2/√(8)) η_0=2.77 and α_2=(2π)^3/2η_0 η_2=0.25.






The data  are modeled by numerically integrating eqs. <ref> and <ref> using four fit parameters, c_0, δU_0, and ϵ, given in the figure captions, and κ_T, which is discussed below. These parameters are extracted by minimizing χ^2 in the central region of the data away from the less dense edges.  The fits are done one parameter at a time across all frequencies for a global best fit, with  c_P, c_V, and η determined by θ_0(c_0). This process is repeated until variation in the parameters no longer results in improvement.  The sensitivity to ϵ is greatest where the  density response shows periodic modulation, while c_0 is dominant in the shape of the leading edges,  Figs. <ref> and <ref>. The fitted δU_0 values are consistent with the value 0.2 ϵ_F0 estimated from the expected modulation depth and the maximum box potential 4.5 ϵ_F0 <cit.>. Blurring arising from the imaging resolution ≃ 3.5 μm, causes the fitted ϵ for the 19 μm data to be smaller than for the 30 μm data. We find that the model captures both the amplitudes and shapes of the density response δ n(z,t)/n_0 for all of the frequencies, Figs. <ref> and <ref>.

From the χ^2 fits for both λ = 19 μm and for λ=30 μm, we obtain c_0=1.30 cm/s, consistent with sound speed measurements in the uniform cloud, which gives 1.40 cm/s. The measured c_0 determines θ_0=0.50 <cit.>.  The temperature was not further increased, because the box potential was not strong enough to confine the gas at significantly higher temperature.

We see that the quality of fits decreases as the speed approaches the adiabatic sound speed, v/c_0=0.88, Fig. <ref> (d).  In the supersonic regime, Fig. <ref>, we find that the fit of the linear hydrodynamic model to the density response is poor, and the thermal conductivity cannot be reliably extracted from the model for any perturbation moving faster than the adiabatic sound speed. We estimate that the hydrodynamic relaxation time is τ=0.13 ms <cit.>, which is fast compared to the period of 1.25 ms at the frequency f=800 Hz used to observe the supersonic response. However, in the supersonic regime, it is possible that the increasing density gradients produce weak shock waves, which are not included in our model.

Sensitivity to κ_T is enabled by measurement at subsonic speeds, as
the frequency v/λ can be less than the relaxation rate γ_κ=κ_T q^2/(n_0c_V). Using eq. <ref>, with θ_0=0.50, we find γ_κ=2π× 760 Hz for λ=19 μm and γ_κ=2π× 305 Hz at λ=30 μm. The fits to the trailing edge of the leading peak rise more sharply for larger κ_T, because the density response propagates closer to the isothermal sound speed c_T<c_0 for large γ_κ and lags behind the leading peak to cause a larger disturbance. From the fits to the subsonic data, we find κ_T=1.14(17)×(15/4)(k_B/m)ħ n_0 at θ_0=0.50.

The fitted thermal conductivity at θ_0=0.50 for the unitary Fermi gas can be compared with the variational calculations <cit.>. As noted above, the high temperature limit of the shear viscosity fits the expansion data down to temperatures just above the superfluid transition. For this reason, we compare the fit value of κ_T/η to the predicted high temperature ratio,  eq. <ref>, κ_T/η=(15/4)(k_B/m). This ratio holds for the unitary gas and for an energy-independent s-wave scattering cross section <cit.>, and is identical to the predictions and measurements for rare gases in the Boltzmann limit <cit.>.  With the viscosity from the expansion data, as used in the fits, η = 1.23 ħ n_0 at θ_0=0.5, we find κ_T/η=0.93(14) ×(15/4)(k_B/m), close to the ratio predicted in the high temperature limit. Finally, we determine the Prandtl number,  Pr=(c_P/m) η/κ_T <cit.>. For θ_0=0.5, we find c_P =1.68 k_B from the equation of state <cit.>, yielding Pr = 0.48(8), which can be compared to the high temperature limit Pr=2/3, obtained from eq. <ref> with c_P =5/2 k_B.

In conclusion, we have directly measured the hydrodynamic response of a unitary Fermi gas subject to a moving spatially periodic perturbation. The measured density perturbations validate a linear response model that incorporates the measured box potential, enabling predictions beyond the approximation of an infinite medium. From the low frequency response, we obtain an estimate of the thermal conductivity of the normal fluid that is consistent with recent predictions. Future measurements in improved box potentials will permit studies of the thermal conductivity at higher temperatures, enabling more precise comparison with benchmark variational calculations. Further, this new method will enable measurement of the thermal conductivity and shear viscosity for imbalanced mixtures in nearly uniform gases, where the transport properties are predicted to change <cit.>.

Note added: Shortly after submission of the present paper, a related study appeared on arXiv, measuring universal sound diffusion in a strongly interacting Fermi gas <cit.>.

Primary support for this research is provided by the Physics Divisions of the National Science Foundation (PHY-1705364) and the Air Force Office of Scientific Research (FA9550-16-1-0378). Additional support for the JETlab atom cooling group has been provided by the Physics Division of the Army Research Office (W911NF-14-1-0628) and by the Division of Materials Science and Engineering, the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (DE-SC0008646).

^*Corresponding author: jethoma7@ncsu.edu










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§ SUPPLEMENTAL MATERIAL


In this supplemental material, we derive the hydrodynamic linear response model, including the thermodynamics and the method of determining the forces arising from the box potential.


 §.§ Hydrodynamic linear response for a normal fluid.

Our analysis is simplified for a normal  Fermi gas, which is a single component fluid with a mass density is ρ≡ n m, where n is the total particle density (we assume a 50-50 mixture of two components) and m is the atom mass. ρ(𝐫,t) satisfies the continuity equation,

    ∂_tρ +∂_i(ρ v_i)=0,

where a sum over i=x,y,z is implied. The mass flux (momentum density) is ρ v_i, with v_i(𝐫,t) the velocity field.

The momentum density and corresponding momentum flux ρ v_i v_j obey

    ∂_t (ρ v_i) +∂_j (ρ v_i v_j)=-∂_i p -n ∂_i U+∂_j(η σ_ij),

Here, -∂_i p-n ∂_i U is the force per unit volume arising from the pressure p and the externally applied potential U(𝐫,t). The last term describes the dissipative forces, which arise only from the shear viscosity  η, as the bulk viscosity vanishes for a unitary gas. The shear stress tensor is σ_ij≡∂_iv_j+∂_jv_i-2δ_ij∇·𝐯/3. Taking the divergence of eq. <ref>, and using eq. <ref>, we immediately obtain

    -∂_t^2ρ+∂_i∂_j (ρ v_i v_j)=-∂_i^2 p-∂_i(n ∂_iU)+∂_i∂_j(η σ_ij).


We are interested in the linear hydrodynamic response to a perturbing external potential δ U(𝐫,t), which leads to first order changes in the density δ n(𝐫,t) and pressure δ p(𝐫,t),

    n(𝐫,t)   =   n_0(𝐫)+δ n(𝐫,t)
    
    p(𝐫,t)   =   p_0(𝐫)+δ p(𝐫,t)
    
    U(𝐫,t)   =   U_0(𝐫)+δ U(𝐫,t).

Here, n_0(𝐫) and p_0(𝐫) are the equilibrium (time independent) density and pressure arising from confinement in the box trap potential, U_0(𝐫). In equilibrium, the velocity field 𝐯_0(𝐫,t)=0 and eq. <ref> requires balance of the forces per unit volume arising from the box trap and the pressure,

    -∇ p_0(𝐫)-n_0(𝐫)∇ U_0(𝐫)=0.

Substituting eq. <ref> into  eq. <ref> and retaining terms to first order in small quantities, we obtain

    ∂_t^2δ n=1/m∇^2 δ p+1/m∇·[n_0(𝐫) ∇δ U+δ n ∇ U_0]+4/3η/n_0 m∇^2∂_tδ n.

Here, the second term on the left side of eq. <ref> is negligible, as the velocity field is first order in small quantities. To obtain the last term in eq. <ref>, we have used the fact that the dissipative force is small compared to the conservative forces and that the density n_0 slowly varies in the region of interest. Then we can ignore the spatial derivatives of η and n_0, taking ∂_i∂_j(η σ_ij)≃η ∂_i∂_jσ_ij=4η/3 ∇^2(∇·𝐯). The velocity field is eliminated using ∇·𝐯≃ -∂_tδ n/n_0, which follows from  eq. <ref>.

We retain the spatial dependence of the driving force per unit volume n_0(𝐫) ∇δ U in eq. <ref>, to assure that it vanishes smoothly at the cloud edges. Here, the perturbing potential δ U is controlled in the experiments. The force arising from the box potential -∇ U_0 is determined from eq. <ref>, using the equation of state p_0[n_0(𝐫),T], where the temperature T is spatially constant in equilibrium. Then,

    ∇ U_0(𝐫)=-(∂ p_0/∂ n_0)_T∇ n_0(𝐫)/n_0(𝐫).

Eq. <ref>, with (∂ p/∂ n)_T =(c_V/c_P) m c_0^2,  yields ∇ U_0(𝐫), from the equilibrium density profile, where c_V and c_P are the heat capacities per particle at constant volume and pressure, and c_0 is the adiabatic sound speed (see Fig. <ref>).

To proceed further, we need an expression for the first order pressure change, δ p, which is determined from eq. <ref> by setting n=n_0 and T=T_0,

    δ p=m c_0^2 δ n+mc_0^2β T_0/c_P n_0 δ s_1.

As discussed in the thermodynamics section below, the first term in eq. <ref> arises from adiabatic compression and in the second term, δ s_1 is the first order change in the entropy per particle, with β is the thermal expansivity. Inserting eq. <ref> into eq. <ref>, we obtain

    ∂_t^2δ n-c_0^2∇^2 δ n-c_0^2 β n_0T_0/c_P∇^2δ s_1-4/3η/n_0 m∇^2∂_tδ n=1/m∇·[n_0(𝐫) ∇δ U+δ n ∇ U_0].


The evolution equation for δ s_1 is determined by the heating rate per unit volume, δq̇,

    T_0(∂_t+𝐯·∇) δ s_1=δq̇/n_0,

where T_0 is the initial, spatially uniform, temperature.
The heating rate arising from the shear viscosity is second order in v_i, which is negligible compared to the heating rate arising from heat conduction. Hence, δq̇≃ -∇·(-κ_T∇δ T)≃κ_T∇^2δ T, where we neglect the spatial derivatives of κ_T. Here, δ T is the first order temperature change, which is determined from eq. <ref> by setting n=n_0 and T=T_0,

    δ T=T_0/c_Vδ s_1 + mc_0^2 βT_0/c_Pδ n/n_0.

Retaining only first order terms in eq. <ref> and using eq. <ref>, we have

    ∂_tδ s_1-κ_T/n_0 c_V∇^2δ s_1=κ_Tmc_0^2/n_0^2c_Pβ ∇^2δ n.


We define δs̃_1≡β T_0 n_0δ s_1/c_P, which has a dimension of density, and take ∇^2→∂_z^2 in eqs. <ref> and <ref> to obtain

    ∂_t^2δ n-c_0^2(∂_z^2 δ n+∂_z^2δs̃_1)-4/3η/n_0 m∂_z^2∂_tδ n=1/m ∂_z[n_0(z) ∂_zδ U+δ n ∂_z U_0].

and

    ∂_tδs̃_1-κ_T/n_0 c_V ∂_z^2δs̃_1=κ_T/n_0c_Vc_P-c_V/c_P ∂_z^2δ n,

where we have used mc_0^2β^2T_0/c_P=(c_P-c_V)/c_V, as shown in below, eq. <ref>.

For the numerical integration, it is convenient to define the dimensionless variables, δñ≡δ n/n_0 and ñ_0(z)≡ n_0(z)/n_0, where n_0 is the average total density. Similarly δs̃_10=δs̃_1/n_0=β T_0 δ s_1/c_P. Further, we define  δŨ(z,t)≡δ U(z,t)/ϵ_F0 and Ũ_0(z)≡ U_0(z)/ϵ_F0, where the local Fermi energy, evaluated for the density n_0, is ϵ_F0=mv_F0^2/2=ħ^2(3π^2n_0)^2/3/(2m). In terms of these variables, eqs. <ref> and <ref> become,

    ∂_t^2δñ-c_0^2(∂_z^2 δñ+∂_z^2δs̃_10)-4/3η/n_0 m∂_z^2∂_tδñ=v_F0^2/2 ∂_z[ñ_0(z) ∂_zδŨ+δñ ∂_zŨ_0]

and

    ∂_tδs̃_10-κ_T/n_0 c_V ∂_z^2δs̃_10=κ_T/n_0c_Vc_P-c_V/c_P ∂_z^2δñ.

We show below how the box force -∂_zŨ_0(z) and the box potential Ũ_0(z) are determined from the measured density profile ñ_0(z).

For the fits, we take the perturbation to have the form,

    δŨ(z,t)=δŨ_0 [1-ϵcos(qz-qvt)] 1 - tanh[(z-v t)/w_z]/2.

In eq. <ref>, δŨ_0 is the strength of the perturbation in units of ϵ_F0 and ϵ is modulation depth in the plane of the atoms. The last factor is unity for z<vt and vanishes for z>vt, where the width w_z (typically a few microns) assures a smooth transition for the numerical integration.




 §.§ Thermodynamics




  §.§.§ Thermodynamic relations


For later use in determining the pressure change δ p  and temperature change δ T, we begin by deriving the well-known formula relating the heat capacity at constant pressure c_P=T(∂ s_1/∂ T)_P to the heat capacity at constant volume c_V=T(∂ s_1/∂ T)_V. In the following, we define the volume per particle V_1=1/n, where n is the local density, and determine the heat capacities per particle from the entropy per particle s_1(p,T).

    c_V=T(∂ s_1/∂ T)_V_1=T(∂ s_1/∂ T)_P+T(∂ s_1/∂ p)_T(∂ p/∂ T)_V_1

Rewriting eq. <ref> using the Maxwell relation (∂ s_1/∂ p)_T=-(∂ V_1/∂ T)_p, obtained from the Gibb's free energy, we have

    c_V=c_P-T(∂ V_1/∂ T)_P(∂ p/∂ T)_V_1.

This result can be written in terms of the thermal expansivity β,

    β≡1/V_1(∂ V_1/∂ T)_P.

Using dp(V_1,T)=0, we have

    (∂ p/∂ T)_V_1=-(∂ p/∂ V_1)_T(∂ V_1/∂ T)_P=
    1/V_1^2(∂ V_1/∂ T)_P(∂ p/∂ n)_T

In the last step, we have used n=1/V_1 so that (∂ p/∂ V_1)_T=-1/V_1^2 (∂ p/∂ n)_T. Inserting eq. <ref>
into eq. <ref> and using eq. <ref>, we obtain the desired relation

    c_V=c_P-β^2T(∂ p/∂ n)_T.


For later use, we rewrite the  Maxwell relation (∂ s_1/∂ p)_T=-(∂ V_1/∂ T)_p as

    (∂ s_1/∂ p)_T=-β V_1

and eq. <ref>, using n=1/V_1 and (∂ p/∂ T)_V_1=(∂ p/∂ T)_n, as

    (∂ p/∂ T)_n=β  n (∂ p/∂ n)_T.


Next, we relate the isothermal sound speed c_T to the adiabatic speed of sound c_0,

    c_0^2≡1/m(∂ p/∂ n)_s_1.


    c_T^2≡1/m(∂ p/∂ n)_T.


Writing p(n,s_1)=p[n,T(n,s_1)], we have

    (∂ p/∂ n)_s_1=(∂ p/∂ n)_T+(∂ p/∂ T)_n(∂ T/∂ n)_s_1.

We rewrite the last factor on the right hand side, using ds_1(p,T)=0  to obtain

    0   =   (∂ s_1/∂ p)_T(∂ p/∂ n)_s_1+(∂ s_1/∂ T)_P(∂ T/∂ n)_s_1
    
    0   =   -β V_1(∂ p/∂ n)_s_1+c_P/T(∂ T/∂ n)_s_1,

where we have used eq. <ref> and the definition of c_P. Hence,

    (∂ T/∂ n)_s_1=V_1β T/c_P(∂ p/∂ n)_s_1.

Inserting eq. <ref> and eq. <ref> into eq. <ref> and using nV_1=1 then yields

    (∂ p/∂ n)_s_1=(∂ p/∂ n)_T+β^2T/c_P(∂ p/∂ n)_T(∂ p/∂ n)_s_1.

With eq. <ref>, eq. <ref> takes the simple form

    (∂ p/∂ n)_s_1=(∂ p/∂ n)_T+c_P-c_V/c_P(∂ p/∂ n)_s_1.

Using eqs. <ref> and <ref> in eq. <ref>, we obtain the well-known result <cit.>

    c_T^2=c_V/c_P c_0^2.

Eqs. <ref> and <ref> then determine β^2 from eq. <ref> in terms of measurable quantities,

    β^2=c_P/c_Vc_P-c_V/m c_0^2 T.


Using the results described above, it is easy to evaluate the pressure change δ p  in terms of the density change δ n and the change in the entropy per particle δ s_1. Writing the pressure as p(n,s_1)=p[n,T(n,s_1)], we have

    δ p   =   (∂ p/∂ n)_s_1δ n+(∂ p/∂ s_1)_nδ s_1
       =   mc_0^2 δ n+(∂ p/∂ T)_n(∂ T/∂ s_1)_nδ s_1.

where we have used eq. <ref>. With eqs. <ref>, <ref>, <ref>, and the heat capacity per particle at constant volume  c_V=T(∂ s_1/∂ T)_n,  we immediately obtain

    δ p=mc_0^2 δ n+mc_0^2 βT/c_P n δ s_1.

For the temperature change δ T, we write T(s_1,n), so that

    δ T=(∂ T/∂ s_1)_nδ s_1+(∂ T/∂ n)_s_1δ n.

Inserting eqs. <ref> and <ref> into eq. <ref> and using V_1=1/n, we find

    δ T=T/c_V δ s_1+mc_0^2 βT/c_Pδ n/n.




  §.§.§ Unitary Fermi gas thermodynamics

For the unitary Fermi gas, universality <cit.> requires that the pressure p and the energy density E are functions only of the density and temperature, related by  p=2  E/3. Dimensional analysis then shows that the energy density takes the simple form

    E=3/5n ϵ_F(n) f_E(θ),

where θ≡ T/T_F is the reduced temperature with T_F the local Fermi temperature. For a balanced 50-50 mixture of two spin components of total density n, the local Fermi energy is  k_BT_F=ϵ_F(n)=mv_F^2/2=ħ^2(3π^2n)^2/3/(2m). The universal function f_E(θ) has been measured by Ku et al., <cit.>, which determines all of the thermodynamic properties. The pressure is then

    p=2/5n ϵ_F(n) f_E(θ).

The entropy density takes a similar form

    s=nk_B f_S(θ)=n s_1(θ),

where f_S(θ) is determined by  f_E(θ).

The adiabatic sound speed eq. <ref> is easily obtained from eq. <ref>, as eq. <ref> requires constant θ for constant s_1,

    c_0^2=1/m(∂ p/∂ n)_θ.

Eq. <ref> immediately gives

    c_0=v_F√(f_E(θ)/3).


Fig. <ref> shows how the adiabatic sound speed c_0 obtained from the measured equation of state varies with θ.

The heat capacity per particle at constant volume takes a simple form. Using eq. <ref>,

    c_V=T/n(∂ s/∂ T)_n=1/n(∂ E/∂ T)_n=k_B3/5f_E'(θ),

where ' denotes differentiation with respect to θ.

We determine heat capacity per particle at constant pressure from eq. <ref> by finding β from eq. <ref>. Using eq. <ref> and <ref>, we have generally

    β =1/n(∂ p/∂ T)_n1/m  c_T^2=c_P/c_V1/mc_0^21/n(∂ p/∂ T)_n.

For the unitary Fermi gas, where p=2  E/3, eq. <ref> shows that (∂ p/∂ T)_n=2 n c_V/3 and

    β_UG=2/3c_P/m c_0^2.

Then,  eq. <ref> reduces to

    c_P-c_V=4/9T/m c_0^2 c_V c_P.

With mc_0^2=2 k_BT_F f_E(θ)/3 from eq. <ref>, eq. <ref> takes the simple form

    c_P-c_V/c_V c_P=1/k_B2/3θ/f_E(θ).

Solving eq. <ref> for c_P, and using c_V from eq. <ref>, then determines

    c_P/c_V=f_E(θ)/f_E(θ)-2/5θ f_E'(θ).



  §.§.§ Determination of the box force


We find the force arising from the confining potential along one axis z, using the measured density profiles n_0(z). We ignore the variation of the density along the line of site and find n_0(z) from the spatially integrated column density, which is obtained from absorption images. Using eq. <ref> with (∂ p/∂ n)_T =(c_V/c_P) m c_0^2 from eq. <ref> and eq. <ref>,

    ∂_z U_0(z)=-m c_0^2[θ(z)] c_V[θ(z)]/c_P[θ(z)]∂_z n_0(z)/n_0(z).

 Here, c_0 is given by eq. <ref> and c_V/c_P is given by eq. <ref>. As ∂_z U_0(z) is independent of the scale of n_0(z), we find ∂_z U_0(z) from the the scaled density ñ_0(z)=n_0(z)/n_0, where
n_0 is the mean central density. Note that the thermodynamic parameters depend only on the reduced temperature θ(z)=θ_0/[ñ_0(z)]^2/3, where we determine θ_0=T_0/T_F(n_0) from the fitted adiabatic sound speed c_0.

For the numerical integration, we express U_0 in units of ϵ_F0, Ũ_0(z)≡ U_0(z)/ϵ_F0 . From eq. <ref>, we have mc_0^2(θ)=mv_F^2 f_E(θ)/3=2 ϵ_F f_E(θ)/3. With ϵ_F/ϵ_F0=[ñ_0(z)]^2/3 and eq. <ref>, we find

    ∂_zŨ_0(z)=-2/3[f_E(θ)-2/5θ f_E'(θ)]∂_zñ_0(z)/[ñ_0(z)]^1/3,

which is also obtained from eq. <ref> by evaluating (∂ p/∂ n)_T directly from eq. <ref>.

To evaluate the derivative ∂_zñ_0(z) on the right hand side of eq. <ref>, we first fit ñ_0(z)=n_0(z)/n_0 with an analytic function,

    f(z)= tanh[(z - z_10)/w_1] -
        tanh[(z - z_20)/w_2]/2 ∑_n a_n z^n.

The difference of the tanh functions produces a top-hat shape of nominal width z_20-z_10 and slopes on each side determined by w_1 and w_2. The flat top is modulated by the multiplying polynomial.  Fig. <ref> shows a typical fit using a fifth order polynomial.

Eq. <ref> then yields the force profile, -∂_zŨ_0(z), Fig. <ref>(a).  The corresponding box potential is then Ũ_0(z)=∫_0^z dz' ∂_z'Ũ_0(z'),  Fig. <ref>(b).




  §.§.§ Hydrodynamic relaxation time


The local shear viscosity can be written as η =τ p <cit.>, where p is the pressure and τ is the hydrodynamic relaxation time. We estimate τ for the unitary Fermi gas using eq. <ref> for p and

    η=ħ n (α_0 θ^3/2+α_2),

with n the local density, α_0=2.77 and α_2=0.25 <cit.>, as discussed in the main paper. Then with τ=η/p,

    τ=5/2τ_Fα_0 θ^3/2+α_2/f_E(θ),

where τ_F=ħ/ϵ_F is the Fermi time. For our experiments, where ϵ_F≃ k_B× 0.15 μK and θ≃ 0.5, we find τ=0.13 ms, which is small compared to the period at the highest frequency of 800 Hz that is employed in the experiments.



