In Situ Thermometry of Fermionic Cold-Atom Quantum Wires
Clément De Daniloff, Marin Tharrault, Cédric Enesa, Christophe Salomon, Frédéric Chevy, Thomas Reimann, Julian Struck
IIn Situ
Thermometry of Fermionic Cold-Atom Quantum Wires
Cl´ement De Daniloff, Marin Tharrault, C´edric Enesa, Christophe Salomon, Fr´ed´eric Chevy, Thomas Reimann, and Julian Struck ∗ Laboratoire Kastler Brossel, ENS-Universit´e PSL, CNRS, SorbonneUniversit´e, Coll`ege de France, 24 rue Lhomond, 75005 Paris, France
We study ensembles of fermionic cold-atom quantum wires with tunable transverse mode pop-ulation and single wire resolution. From in situ density profiles, we determine the temperature ofthe atomic wires in the weakly-interacting limit and reconstruct the underlying potential landscape.By varying atom number and temperature, we control the occupation of the transverse modes andstudy the 1D-3D crossover. In the 1D-limit, we observe an increase of the reduced temperature
T /T F at nearly constant entropy per particle S/Nk B . The ability to probe individual atomic wires in situ paves the way to quantitatively study equilibrium and transport properties of strongly-interacting1D Fermi gases. The 1D world represents an exotic realm of many-bodyphysics. Quantum and thermal fluctuations are enhancedand the dimensional constraint on the motion of par-ticles strongly increases the impact of interactions [1].A paradigm for the resulting unconventional behavior isthe complete collectivization of elementary excitations ingapless 1D systems, known as Tomonaga-Luttinger liq-uids (TLLs) [2–4]. Signatures for TLLs and other char-acteristic 1D states have been observed in a variety ofsolid-state systems, including organic conductors [5, 6],carbon nanotubes [7, 8], semiconductor wires [9–11], an-tiferromagnetic spin chains [12], metallic chains [13] andedge modes of integer and fractional quantum hall states[14]. However, these materials are complex and typicallyfeature uncontrolled inter-dimensional couplings, render-ing quantitative studies difficult.Ultracold atomic gases provide a complementary ap-proach to low-dimensional many-body systems [15–17].Their motional degrees of freedom can be tailored pre-cisely via optical or magnetic potentials, and confine-ment induced resonances provide a means to tune thesign and strength of interactions [18–21]. This high de-gree of controllability makes 1D Fermi gases promis-ing candidates for the observation of elusive phenomena,such as itinerant ferromagnetism [22–24], Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing [25, 26] or Majoranaedge states [27, 28]. So far, only a handful of experi-ments investigated the properties of 1D bulk Fermi gases[20, 29–33], including pioneering works on the control ofinteractions [20, 33] and the effect of spin-imbalance insuch systems [29, 31]. These experiments, however, suf-fered from addressing arrays of wires stacked along twospatial directions with varying atom number, yieldingensemble-averaged measurements. While this drawbackdid not necessarily bias all previous studies, it certainlyprevented progress towards the quantitative understand-ing of many-body problems in 1D. Indeed, as the ther-modynamic state directly depends on the density, theseensemble-averages cover extended regions of the phasediagram, which severely complicates their interpretation and potentially obscures signatures of elusive states. Onan even more fundamental level, ensemble-averages posea problem for the observation of critical behavior or statesthat are characterized by spontaneous pattern formation,e.g., magnetic domains.In this Letter, we report on the preparation, detec-tion and thermodynamic characterization of individualfermionic cold-atom quantum wires in the 1D regime and1D-3D crossover. Our approach relies on the selectiveloading of a single plane of a 2D optical lattice and high-resolution imaging of the resulting single row of atomicwires. This strategy allows us to circumvent line-of-sightaveraging in the absorption images and to directly ac-cess the density distribution of each wire. The ability toresolve 1D density profiles in situ and perform local ther-mometry represents the main result of this work. In ad-dition, we precisely characterize the trapping potential,which is a crucial prerequisite for further thermodynamicstudies of strongly-interacting 1D Fermi gases [34–40].Experimentally, reaching the 1D regime with an atomicgas requires a tight transverse confinement. The occupa-tion of the energetically lowest transverse mode must bepredominant and excitations have to be strongly sup-pressed. This implies that the transverse quantum ofenergy has to be large compared to the energy scalesof the gas, i.e., the Fermi energy E F and thermal en-ergy k B T , were k B is the Boltzmann constant and T the temperature. We employ a large-spacing 2D opticallattice to create an array of independent tube-shapedtraps with the potential V ( ρ, z ) = mω ⊥ ρ / V k ( z ),where ρ = x + y , m is the mass of the atoms and V k ( z ) = mω k z / O ( z ) the axial potential. Giventhis potential, the 1D limit is expressed as k B T (cid:28) ~ ω ⊥ and E F (cid:28) ~ ω ⊥ . The transverse and axial trap frequen-cies are ω ⊥ / π ≈
17 kHz and ω k / π ≈
96 Hz, which cor-responds to a ratio of ω ⊥ /ω k = 177 [41]. The 2D opti-cal lattice is composed of two orthogonally intersectingstanding waves that are each created by interfering a pairof laser beams under a small angle [see Fig. 1(a)]. Thisresults in a lattice constant of d = 2 . µ m = 2 . λ , where a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b k k k k xyz (a) (b) (c)0 20 40 60 80 x (μm)025 z ( μ m ) O D z ( μ m )
35 40 45 x (μm)0.00.3 O D FIG. 1. Individually resolved fermionic quantum wires. (a) Array of tube traps created by superimposing two optical latticesalong the x - and y -direction. The optical lattices are formed by the pairwise interference of the beams under small angles( ] ( k , k ) ≈ ] ( k , k ) ≈ ◦ ), resulting in a lattice spacing of d ≈ . µ m. Here, k i denotes the wave vector of the i th beam.Only a single row of tube traps is populated with atoms. (b) Averaged in situ absorption images of Fermi gases in the 1Dregime. Shown is the optical density (OD). (c) Central region of interest of (b) and the corresponding integrated line profile. λ = 1064 nm is the wavelength of the laser beams. Thelarge lattice spacing renders tunneling between the tubetraps negligible. We measure the in situ density distri-bution of the atomic wires in the tubes through high-resolution absorption imaging [see Fig. 1(b)] [41]. Ourimaging resolution, as defined by the Rayleigh criterion,is 1 . µ m. This is twice lower than the lattice spacing.Therefore, individual atomic wires are fully resolved [seeFig. 1(c),(d)]. Crucially, to avoid line-of-sight integrationalong the imaging axis, only a single row of tube traps ispopulated with atoms [see Fig. 1(a)].Our experiments are conducted with a balanced mix-ture of K atoms in the two energetically lowest hyper-fine states, representing a pseudospin-1 / . -like optical potential [see Fig. 2(a)] [41, 43, 44].At full compression beam intensity, the pancake-shapedpotential is characterized by trap frequency ratios of ω y /ω x = 18 and ω z /ω x = 2 .
6. The position of the pan-cake potential determines which tube traps are selectivelyloaded when ramping up the optical lattice. Once the lat-tice depth is sufficient to inhibit tunneling, the compres-sion potential is removed [see Fig. 2(a)].To verify the loading of a single row of tubes, weimage the atomic cloud along the vertical direction ( z -axis). The large axial spread of the atoms in each tubetrap poses a problem for high-resolution imaging givenits shallow depth of field. We mitigate this issue by us-ing a tomographic optical pumping scheme that transfersatoms outside the central region into undetected hyper-fine states [41, 45, 46]. While it is impossible to resolveindividual atomic wires along this imaging direction, wecan clearly distinguish between the loading of a single anddouble row of the lattice [see insets of Fig. 2(b)]. Trans- lating the compression beam orthogonally with respectto the tube traps, we observe a sharp step-like structurewhen tracking the center of mass of the cloud [see Fig.2(b)]. The plateaus correspond to the population of dif-ferent rows. This indicates the robustness of the loadingprocedure against misalignment of the compression beamand small phase drifts of the optical lattice. In addition,long-term drifts of the lattice are actively compensatedby readjusting the phase at the beginning of each se-quence [41].Now, we turn to the thermodynamic analysis of in situ density profiles. By splitting the absorption images [seeFig. 1(b)] into sections containing only a single atomicwire and integrating over the x -axis, we obtain the in-dividual 1D density distributions n ( z ) per spin state.To improve the signal to noise ratio, we average profileswith comparable atom numbers prepared under the sameexperimental conditions [see Fig. 3(a)]. For the thermom-etry, the 3D scattering length is reduced to | a | = 40 a ,which is still sufficient to establish thermal equilibrium.Here, a is the Bohr radius. To determine the tempera-ture T and chemical potential µ , we use the equation ofstate of the non-interacting Fermi gas n ( µ, T ) = − λ T ∞ X s =0 ( s + 1) Li ( − f s ( µ, T )) , (1)where the sum accounts for the transverse modes withthe energy E s = ~ ω ⊥ ( s + 1) and degeneracy s + 1.Here, Li n ( x ) is the n th order polylogarithm, λ T = h/ √ πmk B T the thermal de Broglie wavelength and f s ( µ, T ) = exp( µ/k B T ) exp( − s ~ ω ⊥ /k B T ) the fugacityof the s th transverse mode. Our assumption of negligi-ble interaction effects in the thermodynamic analysis isnot justified a priori . In fact, in the quantum degenerate1D regime, the ratio of interaction and kinetic energy( E int /E kin ∼ /n ) diverges in the low-density limit.This leads to a non-trivial competition of kinetic energy y ( d )0123456 C o M o ft hea t o m s y ( d ) xz y(a) (I) (II) (III)(b) x (μm) 1300018 y ( μ m ) z (μm)080 y ( μ m ) x (μm) 1300018 y ( μ m ) FIG. 2. Lattice loading procedure. (a) (I) First, the gas (red)is strongly compressed along the y -direction by a repulsiveTEM -like optical potential (green). (II) Second, the opti-cal lattice is ramped up, only populating a single row of thearray of tube traps. (III) Finally, the compression potentialis removed. (b) Center of mass (CoM) of the atoms in thelattice as a function of the compression beam center alongthe y -axis in units of the lattice constant d . The solid line isa guide to the eye. The top inset shows the intensity profileof the compression beam. The insets on the right representcharacteristic images for single- and double-row loading. and interaction effects in the wings of the atomic wires,where the density and degeneracy drop simultaneously.To estimate the influence of interactions on the densityprofiles, we perform an a posteriori consistency checkof our analysis for repulsive and attractive interactions( a = ± a ).Applying the local density approximation µ ( z ) = µ − V k ( z ) to the weakly confined axial direction allows us tofit the equation of state [Eq. (1)] to the measured 1D den-sity distributions. By fitting a set of profiles with varyingtemperatures and atom numbers [see Fig. 3(a)], we ex-tract µ and T as independent parameters for each profileand the reconstructed axial potential as a shared parame-ter for the entire set [47]. More precisely, the anharmonicpart of the potential is modeled by a higher-order poly-nomial while the harmonic part is fixed through an in-dependent measurement [41]. The asymmetry in the po-tential [see Fig. 3(b)] stems from the gravitational force.We observe no significant variation of the axial poten-tial across the 18 central tube traps, which are selectedfor the data analysis. Comparing reconstructed poten-tials from sets with attractive and repulsive interactions (a)(b)(c) 0246810 n D ( μ m − ) N = 606 ± 5N = 476 ± 6N = 355 ± 9N = 277 ± 4N = 179 ± 5N = 118 ± 4N = 75 ± 4 V ∥ ( k H z ) −100 −50 0 50 z (μm)−202 δ V ± ( % ) FIG. 3. (a) Averaged 1D density profiles for individual tubeswith a total atom number N per spin state. The solid linesrepresent fits of the non-interacting equation of state, with T (from top to bottom): 0 . µ K, 0 . µ K, 0 . µ K,0 . µ K, 0 . µ K, 0 . µ K and 0 . µ K. (b) Re-constructed axial potential for a = − a (orange dashed-dotted line) and a = +40 a (blue solid line). The harmonicpart (grey dashed line) of the potential has been indepen-dently measured [41]. (c) Relative difference between the re-constructed potentials for attractive and repulsive interac-tions. reveals only minor differences [see Fig. 3(c)] and thus val-idates the use of the non-interacting equation of state.With the thermometry at hand, the 1D-3D crossover,driven by the gradual occupation of transverse modes,can be precisely characterized. We obtain local dimen-sionless quantities by normalizing the relevant energyscales with the Fermi energy, which is determined by theimplicit equation n = r mE F h j EF ~ ω ⊥ k X s =0 ( s + 1) r − s ~ ω ⊥ E F , (2)where b x c denotes the floor function. This equation sim-plifies to E F = h n / m in the 1D regime. The popula-tion of transverse modes can either be caused by thermalexcitations or arise as a consequence of Pauli blocking.By changing the initial evaporation parameters in thecrossed optical dipole trap prior to loading the atomsinto the lattice, we vary the final atom number and tem-perature in the tubes. This way, we control the transversemode population and can reach the deep 1D regime with k B T . . ~ ω ⊥ and E F . . ~ ω ⊥ [see Fig. 4(a)] [41].The temperature spread across different atomic wires isyet another sign that tunneling between the tube trapsis strongly suppressed, which leads to an early thermaldecoupling during the lattice loading procedure.Previous 1D Fermi gas experiments inferred estimatesof the degeneracy based on thermometry of the initial 3Dcloud before loading the optical lattice. These estimatesrely on the assumption of an isentropic loading proce-dure, which is questionable due to various technical heat-ing processes, e.g, laser intensity noise, and the suppres-sion of thermalizing two-body collisions in the 1D regime[48–50]. Here, we do not rely on this assumption and de-termine in situ the local reduced temperature T /T F andentropy per particle SN k B = P ∞ s =0 ( s + 1) (cid:16) Li ( − f s ) − ln( f s )Li ( − f s ) (cid:17)P ∞ s =0 ( s + 1)Li ( − f s ) . (3)We observe that S/N k B stays nearly constant in the en-tire crossover region, whereas T /T F displays a suddenincrease in the low atom number and temperature limit[see Fig. 4(b),(c)]. This is a clear signature of the 1D-regime, where the equation of state is strongly alteredwith respect to the 3D case. Nevertheless, the system re-mains degenerate ( T /T F <
1) in a large part of the 1Dregime, which enables the study of 1D quantum many-body physics in the future. Wire-resolved measurementsreveal a systematic increase of the entropy per particletowards the central tube, where the density is the high-est [see Fig. 4(d)]. This density dependence speaks foran interaction related effect that influences the latticeloading procedure. It also highlights the importance andrelevance of our thermometry based on individual atomicwires.A direct follow-up study to the work presented hereis the measurement of the equation of state of strongly-interacting fermionic wires. Within the framework of thelocal density approximation, the precise knowledge of theaxial potential will allows us to locally relate the 1D den-sity and chemical potential at any interaction strength.The local pressure and compressibility can then be ob-tained from the integral and derivative of the density withrespect to the chemical potential [38, 39, 51]. From theseobservables, further thermodynamic quantities can be de-termined, such as the reduced temperature
T /T F . The-oretically, the ground state and low-temperature regime( T (cid:28) T F ) of the interacting 1D Fermi gas can be solvedwith the Bethe ansatz [52]. However, for the general fi-nite temperature case, the situation is significantly morechallenging and theoretical studies are sparse [53].The individual probing of 1D Fermi gases promises (a)(b)(c)(d) k B T / ħ ω T / T F S / N k B ⟨ S / N k B ⟩ Tube trap index0.00.20.40.60.80.00.51.01.50.0 0.5 1.0 1.5 2.0E F /ħω ⟂
012 −5 0 51.01.52.0
FIG. 4. Thermodynamics of the 1D-3D crossover. The datapoints correspond to the center of each tube trap ( z = 0),where the Fermi energy is the highest. The colors representa selection of three characteristic tubes. (a) The tempera-ture normalized by the transverse frequency k B T / ~ ω ⊥ and(b) the reduced temperature T /T F . The grey gradient in (a)depicts the 1D-limit. The solid lines in (b), emanating fromthe data points, represent the continuous change in E F alongeach atomic wire. (c) The entropy per particle S/Nk B . (d)Mean value of the data shown in (c). further insight into elusive states of matter and criti-cal behavior. This includes the observation of the highlysought FFLO phase [25, 26, 29] and the study of TLLsfeaturing collectivized excitations and spin-charge sepa-ration [32, 54–57]. Of particular interest is the interplayof strong interactions and the suppression of thermaliz-ing collisions in 1D [48, 50, 58], which strongly impactsout-of-equilibrium and transport phenomena.We thank Tarik Yefsah for stimulating and helpful dis-cussions and critical reading of the manuscript as well asAntoine Heidmann for hosting the project at LaboratoireKastler Brossel and his active support for the completionof this work. J.S. was supported by LabEX ENS-ICFP:ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. Weacknowledge support from DIM Sirteq (Grant EML19002465 1DFG) and Fondation S. et C. del Duca (Grant61846).C.D. and M.T. conducted the measurements and dataanalysis. C.D. and J.S. performed the theoretical com-putations. C.D., M.T., C.E., T.R. and J.S. designedand constructed the relevant parts of the experimentalapparatus for the project. J.S. conceptualized, plannedand supervised the project. C.D. and J.S. prepared themanuscript with comments from all authors. ∗ Corresponding author: [email protected][1] T. Giamarchi,
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Thermometry of Fermionic Cold-Atom Quantum Wires
Cl´ement De Daniloff, Marin Tharrault, C´edric Enesa, Christophe Salomon, Fr´ed´eric Chevy, Thomas Reimann, and Julian Struck ∗ Laboratoire Kastler Brossel, ENS-Universit´e PSL, CNRS, SorbonneUniversit´e, Coll`ege de France, 24 rue Lhomond, 75005 Paris, France
LARGE-SPACING OPTICAL LATTICE
The 2D optical lattice consists of two superim-posed standing waves, which are each created by in-terfering a pair of laser beams under an angle of2 θ ≈ ◦ . The standing waves along the x - and y -axis are formed by beam pairs with the wavevec-tors k , = k ( ± sin( θ ) , , − cos( θ )) and k , = k (0 , ∓ sin( θ ) , − cos( θ )), respectively. Here, k = 2 π/λ isthe wavenumber and λ = 1064 nm. The waists of the lat-tice beams are approximately 180 µ m, which results in avariation of ω ⊥ of less than 2% across the 18 central tubetraps that are used for data analysis. We collect all fourlattice beams in a microscope objective after they havepassed the atomic cloud and image them onto a CCDcamera [see Fig. S1]. This gives us an absolute phase ref-erence of the lattice. Through piezo-driven mirrors in thebeam paths [S1], we control the phase and compensatelong-term drifts at the beginning of each experimentalsequence.The intensity ramps of the compression beam and op- (a) (b)y xz k k k k y (μm)050 x ( μ m ) FIG. S1. Direct imaging of the optical lattice. (a) The laserbeam pairs (blue and red) forming the lattice are collected bya microscope objective and (b) refocused onto a CCD camera.For the sake of completeness, the crossed optical dipole trapbeams (yellow) are also included in (a). N o r m . i n t en s i t y −1.5−1.2−0.9−0.6−0.30.0 a ( ) (I) (II) (III) FIG. S2. Lattice loading ramps. (Green dashed line) Normal-ized intensity of the compression beam. (Orange solid line)Normalized intensity of the lattice laser beams. (Blue dashed-dotted line) s -wave scattering length. The roman numberscorrespond to the schematic depiction in figure 2(a). tical lattice for the preparation of a single row of tubesare shown in figure S2. Also depicted is the change of the s -wave scattering length during the loading procedure,starting from a = − a in the crossed optical dipoletrap for the initial evaporative cooling of the atomic gas. ABSORPTION IMAGING
We resonantly image the energetically lowest hyper-fine state | F = 9 / , m F = − / i at a magnetic bias fieldabove 200 G on the D2 cycling transition. Prior to imag-ing, all optical potentials are switched off in order toavoid spatially dependent AC Stark energy shifts. Ashort imaging pulse of t img = 12 µ s ensures that thegas expands only marginally in the transverse direction( ω ⊥ t img = 0 .
2) and that the frequency shift due to theDoppler effect is negligible. We have verified the latter bymeasuring the velocity of the atomic cloud after imagingpulses of different durations [S2]. A linear increase of thevelocity with the pulse duration implies a constant forceon the atoms during the imaging process and hence nosignificant frequency shift. The pulse duration of 12 µ slies well within this regime.To determine the column density ˜ n of the atomic gasalong the probe beam direction, we use the extendedBeer-Lambert law σ ˜ n = ln( I ref /I abs ) + ( I ref − I abs ) /I sat , (S1) a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b (a)(b) (c) z y x x (μm)04080 0.00.8 O D x (μm)04080 z ( μ m ) FIG. S3. Optical pumping tomography. (a) Imaging systemprojecting the shadow of an opaque slit onto the row of atomicwires. (b) and (c) correspond to absorption images of theatomic wires without and with optical pumping, respectively. which accounts for the saturation of the atomic transi-tion. Here, σ is the absorption cross section, I sat thesaturation intensity, I ref and I abs the imaging intensitiesbefore and after the atoms, respectively. Typically, we op-erate with a saturation parameter of s = I ref /I sat ≈ − < .
5) under the same experimental conditions andvary the probe beam power. Subsequently, we find thesaturation parameter that yields a constant column den-sity [see eq. (S1)] for varying probe beam powers. Theestimated systematic uncertainty on the saturation pa-rameter is 10%.
OPTICAL PUMPING TOMOGRAPHY
The large axial extent of the atomic wires poses a prob-lem for the z -axis imaging system [see Fig. S2(a)], whichfeatures a depth of field of ± µ m. We mitigate this is-sue by optically pumping atoms located outside of a cen-tral slice into the upper hyperfine ground state manifold( m J = 1 / µ m, while the resolu-tion of the imaging system is 5 µ m. This is consistent withthe measured 4 µ m full width at half maximum of theatomic slice after the tomography [see Fig. S3(c)]. In theexperiment, the pumping pulse illuminates the atomiccloud for 1 µ s before the absorption image is taken. (a)(b)(c) 949698100 ω ∥ / π ( H z ) −5 0 5Tube trap index4.04.24.4 | Δ z | ( μ m ) −4 0 4 z (μm)−101 δ V ha r m ( % ) FIG. S4. Gradient measurement of the axial frequency. (a)Axial frequency of each tube trap. (b) Corresponding dis-placement | ∆ z | of the maximum of the 1D density distri-bution. The dashed lines in (a) and (b) indicate the meanvalue over all tubes. (c) Relative difference δV harm betweenthe harmonic approximation and the fully reconstructed ax-ial potential. MEASUREMENT OF THE AXIAL FREQUENCY
Due to the anharmonicity of the potential and thelarge axial extent of the atomic wires, a reliable mea-surement of the axial frequency via induced oscillationsis not feasible. Instead, we apply calibrated magneticfield gradients ± B z along the axial direction and mea-sure the differential displacement 2∆ z of the maximumof the 1D density distribution. In the small displacementlimit (∆ z → ω k = p µB z /m ∆ z , where µ is the magnetic moment.We observe no statistically significant shift of the mea-sured frequency across the 18 central tube traps [see Fig.S4]. Typically, the displacement from the center of thetrap is ∆ z ≈ ± µ m. A self-consistency check with thefully reconstructed potential demonstrates that the rela-tive deviation from a harmonic potential in this region isat most 1% [see Fig. S4(c)]. EQUATION OF STATE FITTING PROCEDURE
Using the local density approximation µ ( z ) = µ − V k ( z ), with µ = µ ( z = 0), we simultaneously fit a largenumber of 1D density profiles using the non-interactingequation of state n ( µ ( z ) , T ) [Eq. (1)]. In order to accountfor the anharmonicity of the axial potential, we use a 7thorder polynomial model V k ( z ) = 12 mω k z + X p =3 a p z p , (S2)3 E F /ħω ⟂ η ( μ , T ) FIG. S5. Relative occupation η of the lowest transverse modeat the center of the tube traps ( z = 0), where η is the small-est. Different colors represent three characteristic tube traps.The data points correspond to the ones shown in Fig. 4. where the axial frequency ω k is directly measured as de-scribed in the previous section and the higher-order coef-ficients a p are left as fit parameters. We have verified thatthe reconstructed potential in the region with atoms isnot changed for polynomial models of higher order than7. To prepare the 1D density profiles for the fitting proce-dure, we first sort all absorption images of a given data setaccording to their total atom number. Subsequently, thislist is grouped in intervals of 10 images, which are thenaveraged. From the resulting images, the 1D density pro-files for each tube are extracted. For example, the entiredata set for the s -wave scattering length of a = +40 a consists of 270 absorption images. After averaging andselecting the 18 central tubes, we end up with a set of 486 individual 1D density profiles.To measure T and µ for a given profile and to deter-mine the polynomial coefficients a p [Eq. (S2)], we simul-taneously fit an entire set of profiles. For each individual1D profile, β = 1 /k B T and βµ constitute independent fitparameters, while the coefficients a p serve as shared pa-rameters for the whole set. Statistical errors of the ther-modynamic quantities are obtained from the covariancematrix of the fit. TRANSVERSE MODE OCCUPATION
The transverse mode population of the tube traps is ameasure of the dimensional character of the atomic wires.The relative occupation of the lowest mode η ( µ, T ) = Li ( − f ( µ, T )) P ∞ s =0 ( s + 1) Li ( − f s ( µ, T )) , (S3)is given by the equation of state [Eq. (1)]. In the lowatom number and temperature limit, η saturates to one[see Fig. S5], which, by definition, corresponds to the 1Dregime. ∗ Corresponding author: [email protected][S1] E. Magnan, J. Maslek, C. Bracamontes, A. Restelli,T. Boulier, and J. V. Porto, A low-steering piezo-drivenmirror, Rev. Sci. Instrum. , 073110 (2018).[S2] K. Hueck, N. Luick, L. Sobirey, J. Siegl, T. Lompe,H. Moritz, L. W. Clark, and C. Chin, Calibrating highintensity absorption imaging of ultracold atoms, Opt. Ex-press25