Probing quantum and thermal noise in an interacting many-body system
S. Hofferberth, I. Lesanovsky, T. Schumm, A. Imambekov, V. Gritsev, E. Demler, J. Schmiedmayer
aa r X i v : . [ c ond - m a t . o t h e r] M a r Probing quantum and thermal noise in an interacting many-body system
S. Hofferberth,
1, 2, 3
I. Lesanovsky,
2, 4
T. Schumm, A. Imambekov,
3, 5
V. Gritsev, E. Demler, and J. Schmiedmayer
1, 2 Atominstitut der ¨Osterreichischen Universit¨aten, TU-Wien, Stadionallee 2, 1020 Vienna, Austria Physikalisches Institut, Universit¨at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany Department of Physics, Harvard University, Cambridge, MA 02138, USA Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria Department of Physics, Yale University, New Haven, CT 06520,USA (Dated: October 22, 2018)The probabilistic character of the measurement process is one of the most puzzling and fascinatingaspects of quantum mechanics. In many-body systems quantum mechanical noise reveals non-localcorrelations of the underlying many-body states. Here, we provide a complete experimental anal-ysis of the shot-to-shot variations of interference fringe contrast for pairs of independently createdone-dimensional Bose condensates. Analyzing different system sizes we observe the crossover fromthermal to quantum noise, reflected in a characteristic change in the distribution functions fromPoissonian to Gumbel-type, in excellent agreement with theoretical predictions based on the Lut-tinger liquid formalism. We present the first experimental observation of quasi long-range order inone-dimensional atomic condensates, which is a hallmark of quantum fluctuations in one-dimensionalsystems. Furthermore, our experiments constitute the first analysis of the full distribution of quan-tum noise in an interacting many-body system.
I. INTRODUCTION
The probabilistic nature of Schr¨odinger wave functionsand the uncertainty principle are crucial aspects of themodern understanding of quantum matter. Starting withthe famous Bohr-Einstein debates, intrinsic quantum me-chanical noise has been the subject of numerous discus-sions and controversies [1]. The analysis of quantum andthermal noise is not only important in the study of fun-damental problems in physics but also crucial for under-standing the ultimate limits of optical and electric detec-tors, sensors, and sources.The understanding of noise in the non-classical statesof light facilitated the development of the theory of pho-todetection and led to the foundation of quantum op-tics [2]. In solid state systems current fluctuations wereused to probe the nature of electrical transport in meso-scopic electron systems [3, 4] and to investigate quantumcorrelations and entanglement in electron interferometers[5, 6].In atomic physics, noise correlation analysis [7], wasemployed to study quantum states in optical lattices[8, 9], pair correlations in fermi gases [10], the countingstatistics in an atom laser [11], and the Hanburry-Brown-Twiss effect for both bosons and fermions [12, 13].Interference experiments provide a different powerfultool which allowed, for example, the study of macro-scopic phase coherence [14], critical fluctuations [15],thermal fluctuations in elongated condensates [16], andthe Berezinskii-Kosterlitz-Thouless transition in a two-dimensional quantum gas [17]. Recently it has been sug-gested that the full statistics of fluctuations in the con-trast of interference fringes can be used to probe highorder correlation functions and reveal non-trivial phasesof low-dimensional condensates [18, 19].In this paper we develop this idea further and provide a complete experimental analysis of the shot-to-shot vari-ations of the interference fringe contrast for pairs of in-dependently created one-dimensional Bose condensates.For long system sizes we find that both the average con-trast and its variations are dominated by thermal fluc-tuations. For smaller systems we demonstrate that thedistribution functions of fringe contrast provide unam-biguous signatures of quantum fluctuations. Earlier ex-periments with one-dimensional condensates in opticallattices observed manifestations of strong atomic inter-actions in the three body recombination rate [20], the to-tal energy [21], and the momentum distribution of atoms[22]. However the power-law nature of the correlationfunctions, which is the hallmark of one-dimensional quasicondensates, has never been observed before. Our exper-iments provide the first experimental demonstration ofquasi long-range order in one-dimensional condensates.This is the smoking gun signature of quantum fluctua-tions in systems of weakly interacting ultracold atoms.More importantly, our work constitutes the first mea-surements of the full distribution functions of quantumnoise in an interacting many-body system.
II. EXPERIMENTAL PROCEDURE
Our experiments are performed using two indepen-dent one-dimensional quantum degenerate atomic Bosegases created in a radio-frequency-induced micro-trap[23, 24, 25] on an atom chip [26, 27]. Each sample con-tains typically 3000 − T and chemical potential µ fulfill-ing k B T, µ < hν ⊥ , where ν ⊥ = 3 . FIG. 1: Experimental setup and observed interference pat-terns. (a)
Two independent 1d Bose gases are created by firstsplitting a single highly elongated magnetic trap on an atomchip holding a thermal ensemble of atoms into a double-wellusing radio-frequency-induced potentials. In a second step theseparate parts are evaporatively cooled to degeneracy, pro-ducing two individual 1d condensates (schematic on the left).The two systems are then simultaneously released from thetrapping potential and the resulting interference pattern isrecorded with standard absorption imaging. The vertical ori-entation of the initial system is chosen so that the interferencepattern can be imaged along its transverse direction, parallelto the atom chip (illustration on the right side). (b)
Colorcoded images of the resulting density patterns. The observedinterference fringes show a meandering over the length of thesystem (z-direction), which is due to the local differences inrelative phase between the two original 1d condensates. Con-sequently, this waviness of the patterns contains informationabout the phase correlations in the individual condensates. pattern is recorded using standard absorption imaging(figure 1a).Examples of the observed fringe patterns are shown infigure 1b. While the interference patterns have high lo-cal contrast, the interference fringes as a whole are notstraight lines. This meandering character of the inter-ference patterns shows that the relative phase betweenthe two condensates is not constant but fluctuates frompoint to point. These phase variations originate fromboth quantum and thermal fluctuations in the original 1dcondensates and reflect the non-mean-field character oflow-dimensional systems. Integrating local interferencepatterns over a finite length L leads to summing inter-ference patterns which are not in phase with each otherand results in a reduction of the total fringe contrast (fig-ure 2). This reduction of the interference contrast, andits statistical fluctuations, contains important informa-tion about the phase correlations of the individual 1dcondensates and is the main quantity addressed in this FIG. 2: Analysis of the observed interference patterns. Forquantitative analysis, we integrate over central slices of vary-ing length L of the density profiles in the longitudinal di-rection as indicated by the shading in the top row, to ob-tain multiple transverse line density profiles. We then extractthe interference amplitude | A Q | by Fourier transforming theseprofiles and extracting the Fourier coefficient correspondingto the fringe spacing Q . To illustrate contrast reduction withincreasing L , the fringe patterns shown in the bottom row arenormalized. Modulated cosine fits to these profiles then yieldcontrasts C ( L ) which decrease with L . Note that the inter-ference amplitude | A Q | as defined in equation (1) is relatedto this contrast as | A Q | = n d L × C . Consequently, as canbe seen from equation (3), | A Q ( L ) | increases with L , whereasthe contrast C ( L ) decreases with L . work. III. THEORETICAL MODEL
Before we proceed to the statistical analysis of the ex-perimental data we present a quick overview of the the-oretical foundations of our study. The analysis is per-formed in two steps. In the first step we analyze the average amplitude of interference fringes as a function ofthe integration length L , the second step then analyzesthe quantum and thermal noise contributions to the shot-to-shot fluctuations in the average contrast Our interference patterns show a periodic density mod-ulation at the interference wave vector Q = md/ ~ t , where m is the mass of the atoms, d is the in-trap separation ofthe two 1d systems, and t is the expansion time. Assum-ing ballistic expansion, the complex amplitude of thisdensity modulation after integration over a length L isgiven by [18, 30] A Q ( L ) = Z L/ − L/ dz a † ( z ) a ( z ) . (1)Here a , are the boson annihilation operators within thetwo original one-dimensional condensates before the ex-pansion. The phase of A Q ( L ) describes the position of in-terference fringes and is determined by the relative phasebetween the two condensates averaged between -L/2 and+L/2 .Since we perform interference experiments with inde-pendent condensates, the phase of A Q is random, andthe expectation value h A Q i is zero. This does not implythe absence of fringes but shows the unpredictable ran-dom phase in individual interference patterns [31, 32, 33].Consequently, to study the contrast statistics of theinterference patterns one has to consider the quantity h| A Q ( L ) | i . This quantity is independent of the overallphase difference but is strongly affected by phase twistingwithin each condensate.In the case of ideal non-fluctuating condensates oneexpects to find perfect contrast for any size of the system.This implies h| A Q ( L ) | i ∝ L . In the opposite regimeof short range phase correlations with finite correlationlength ξ φ , the net interference pattern comes from addingup fringes in L/ξ φ uncorrelated domains. In this casethe net interference pattern is strongly suppressed andappears only as a square root fluctuation, h| A Q ( L ) | i ∝ Lξ φ .More precisely, h| A Q ( L ) | i is determined by the inte-gral of the two point correlation function: h| A Q ( L ) | i = Z L/ − L/ dz Z L/ − L/ dz (2) h a † ( z ) a ( z ) ih a † ( z ) a ( z ) i . A special feature of one-dimensional systems of inter-acting bosons is the dramatic enhancement of fluctua-tions. Even at T = 0, true long-range order is not possi-ble and only quasi-condensates with a power-law decay ofthe correlation function h a † ( z ) a ( z ) i exist [34]. At finitetemperatures one finds exponential decay of the correla-tion function for distances | z − z | exceeding a thermalcorrelation length ξ φ ( T ) [28].To adequately describe these systems, a beyond mean-field theory is required. A powerful non-perturbativeapproach which describes the long distance behaviorof the correlation functions of one-dimensional systemsis the Luttinger liquid theory (see methods and Refs.[35, 36, 37]), on which we base our further analysis.Using a standard expression for the two point correla-tion function in Luttinger liquid theory we obtain h| A Q ( L ) | i = C ξ h L + C (cid:18) Lξ h (cid:19) − /K f ( ξ φ ( T ) L , K ) . (3)Here, K ≈ π ~ q n d gm is the Luttinger parameter for theweakly interacting 1d Bose gas, with n being the 1d linedensity, g = 2 hν ⊥ a s the effective 1d coupling constant,and a s being the s-wave scattering length. ξ h = ~ √ mgn d is the healing length and ξ φ ( T ) = ~ n d πmk B T is the thermalcorrelation length of the 1d condensates (for the weaklyinteracting regime), while C and C are numerical con-stants of order unity. The function f ( x, K ) is given by f ( x, K ) = Z Z dudv πx sinh( π | u − v | x ) ! /K . (4)Note that the finite number of particles can in principlelead to corrections to equation (4) [30, 38]. We checked that for our parameters this shot noise is of no impor-tance even for the smallest L we investigate.The first term in equation (3) is generally small. Itdescribes contributions from the short distance part ofthe correlation function, which is not sensitive to ther-mal fluctuations (provided that T < µ as in our system)or the non-trivial effects of quantum fluctuations in 1dsystems. The second term shows how quantum and ther-mal fluctuations in the 1d condensates affect h| A Q ( L ) | i .Let us first discuss the case of low temperatures and/orshort system sizes ( L/ξ φ ( T ) ≤ h| A Q ( L ) | i ∝ L . In the other extreme, impenetrablebosons (Tonks-Girardeau gas) have very strong fluctua-tions and their interference pattern corresponds to shortrange correlations, h| A Q ( L ) | i ∝ L (see discussion above,eq. (2)). For finite interaction strength we find somethingin between, resulting in the scaling h| A Q ( L ) | i ∝ L − /K .Finite temperature introduces thermal fluctuations,which create phase fluctuations with a temperature de-pendent correlation length ξ φ ( T ). When L > ξ φ ( T ) ther-mal fluctuations dominate and the interference amplitudescales as h| A Q ( L ) | i ∝ Lξ φ ( T ).The experimentally observed interference patterns pro-vide us with more information than just the averagevalue h| A Q ( L ) | i . As a second step of our analysis weconsider the shot-to-shot fluctuations of individual mea-surements, which are characterized by the higher mo-ments h| A Q | n i and ultimately by the entire distribu-tion function W ( | A Q ( L ) | ). For visualizing the shot-to-shot fluctuations of the interference amplitude, it turnsout to be convenient to consider the normalized variable α ( L ) = | A Q ( L ) | / h| A Q ( L ) | i and its distribution func-tion W ( α ( L )). The importance of the higher moments h| A Q | n i is that they are directly related to the higherorder correlation functions of the 1d interacting Bose gas[18].In the following, we only summarize the scaling of thedistribution function, a formal theoretical approach isdiscussed in the methods. In the case of non-interactingideal condensates, i.e. perfect interference patterns ineach measurement, the distribution function W ( α ( L ))approaches a delta function. When interactions are weakbut finite, we expect a narrow distribution of width 1 /K and W ( α ( L )) to approach a universal Gumbel distri-bution [39]. In the limit of long integration lengths, L ≫ ξ φ ( T ), thermal fluctuations dominate. As discussedabove, in this case the net interference pattern comesfrom adding local interference patterns from many un-correlated domains resulting in the distribution functionbeing Poissonian. For integration lengths comparable to ξ φ ( T ) both quantum and thermal fluctuations are impor-tant. In this regime we expect W ( α ( L )) to show a doublepeak structure, with the peak at small amplitudes comingfrom the thermal noise and the peak at finite amplitude
10 20 30 40 500246 x 10 L [ µ m] < | A Q | > T = 31 nKT = 47 nKT = 65 nK
FIG. 3: Length dependence of the average contrast. The datapoints show the measured h| A Q | i for three different temper-atures T . Error bars indicate s.e.m. Each data point contains50 individual interference measurements. The solid lines arefits of equation (3) to the data with T as free parameter (seemethods) and parameters C = 0 .
91 and C = 0 . K and ξ h are determined independently from measurements of n d and ν ⊥ . from quantum noise. IV. ANALYSIS OF THE AVERAGEINTERFERENCE AMPLITUDE
We now turn to the analysis of our experimental data,starting with the average interference amplitude square h| A Q | i and its variation with system size L .To extract | A Q | from the observed interference pat-terns, we first obtain transverse density profiles inte-grated over the longitudinal direction for different length L , as shown in figure 2. We then fit a cosine functionwith a Gaussian envelope to the resulting fringe profilesto extract the relative phase and the interfering ampli-tude | A Q | as functions of L (see methods). To ensurehomogeneous 1d density, we restrict our analysis to thecentral 50% of the system, where n d ( z ) varies at mostby ∼
10% for the largest L considered. This modula-tion is neglected and we obtain a single value for n d byaveraging over the atomic density in this center region.Figure 3 shows the experimentally observed averageinterference amplitude squares h| A Q | i for three differenttemperatures, with the density n d = 50 µ m − and thetransverse trapping frequency ν ⊥ = 3 . K = 42 and healing length ξ h = 0 . µ m) iden-tical for all three data sets. The higher temperature datasets are obtained by waiting for different times after theinitial preparation of the two condensates. During thiswaiting time, the system heats due to residual noise inthe magnetic trapping fields.To compare measurement and theory, we fit the func- tion (3) to the experimental data (figure 3) with the tem-perature T as a free parameter (see methods). We findthe functional behavior of the measured contrasts to bein very good agreement with the theoretical predictions.This is of particular interest as the shape of these curvesis determined by both the quantum and thermal contri-butions to the average contrast, as discussed above. Forintegration length longer than 20-30 µ m we observe a lin-ear dependence of h| A Q ( L ) | i on L . This corresponds tothe L >> ξ φ ( T ) regime where thermal fluctuations dom-inate.For shorter segment lengths, quantum fluctuations areimportant. However the analysis presented in figure 3 isnot sufficient to make the case for quantum fluctuations.The Luttinger parameter for our system is K = 42, andit is impossible to observe the L − /K correction to theideal case (noise-free) power law L in the limited rangeof lengths available. From figure 3 we can not prove thatfluctuations are present at all for such short system sizes.We will address this in the next part of our analysis bydemonstrating that quantum fluctuations manifest un-ambiguously in the shot-to-shot fluctuations of | A Q ( L ) | ,rather than in the h| A Q ( L ) | i average value.¿From the fits we obtain the temperatures T =31 , ,
65 nK for 0 , ,
100 ms waiting time, respectively.As expected, the greater the waiting time, the higherthe temperature of the system. We note here that thismethod measures the temperature of collective excitationin the condensate. We cannot confirm that this temper-ature is identical to that of the residual thermal atoms inthe trap. Reliable detection of the thermal backgroundis possible only down to T ≈
80 nK in our setup.The contrast method we present here can be used tomeasure the temperature of collective modes of 1d Bosegases at extremely low temperatures and small atomnumbers, suggesting the usefulness of this method forprecise thermometry of 1d condensates when conven-tional methods fail.
V. ANALYSIS OF THE FULL DISTRIBUTIONFUNCTIONS OF INTERFERENCE AMPLITUDE
We now analyze the full information contained in thestatistics of the interference contrast.In figure 4, we show histograms of the measured dis-tributions W ( α ( L )) for four different length scales L andtwo different temperatures, T = 30 nK (upper row, fig-ure 4(a)) and T = 60 nK (lower row, figure 4(b)), ob-tained using the contrast average method discussed inthe previous section. Both data sets were obtained from1d condensates with n d = 60 µ m − , ν ⊥ = 3 . K = 46.For a detailed comparison between measurement andtheory, we numerically calculate the distribution func-tions for the corresponding experimental parameters foreach histogram (see methods and Ref. [40]). We empha-size that once we know the temperature of the system, F=4.1 F=1.7 F=1.1 F=0.8F=2.1 F=0.9 F=0.6 F=0.4
FIG. 4: Distribution functions of the measured in-terference contrasts for different lengths L . (a) Thelength-dependent normalized interference contrasts α ( L ) = | A Q ( L ) | / h| A Q ( L ) | i of 170 individual experimental realiza-tions with identical parameters ( n d = 60 µ m − , ν ⊥ =3 . K = 46) are displayed as histograms. The redcurves show the corresponding calculated distributions for T = 30 nK ( ξ φ ( T ) = 4 . µ m). (b) Histograms of 200 individ-ual measurements with the same parameters as in (a) , buthigher temperature T = 60 nK ( ξ φ ( T ) = 2 . µ m). For bothsets we observe very good agreement between experiment andtheory. In particular, the predicted change of overall shapeof the distribution functions from single peak to Poissonianwith decreasing F = ξ φ ( T ) /L (increasing L and T ) is verywell reproduced by the experimental data. there are no free parameters remaining in this analysis. Itis truly remarkable that the experimentally measured dis-tribution functions are in such excellent agreement withthe theoretical prediction (figure 4). In particular weclearly observe the transition from the regime dominatedby quantum fluctuations for small L at low temperatureto the one dominated by thermal fluctuations for large Land higher temperature.More quantitatively, the shape of the distribution func-tions is determined by a single dimensionless parameter F = Kξ T /L . For large F , accessed either at low temper-ature T or small system length L we find a single asym-metric peak in the distribution function W ( α ), resultingfrom a universal extreme-value statistics Gumbel distri-bution [39] which is a smoking gun signature of quantumfluctuations and the power-law behavior of the correla-tion functions.The Gumbel distribution has a characteristic asym-metric shape and typically appears when describing rareevents such as stock market crashes or earthquakes,which go predominantly in one direction. This suggeststhat suppression of the contrast of interference fringesdue to quantum fluctuations is dominated by rare butstrong fluctuations of the phase of the bosonic fields a , ( z ). Most of the time we find very small phase me- andering which does not affect the contrast significantly.Only occasionally there is a strong fluctuation leading toa noticeable decrease of the interference fringe contrast.For longer system sizes L and higher temperatures T ,we observe that the distribution functions become Poisso-nian, characteristic for the dominance of thermal fluctu-ations and the exponentially decaying correlations. Thesystem can be thought of as consisting of domains of size ξ φ ( T ), with uncorrelated phases in each of the domains.In this case, adding up interference amplitudes (complexnumbers) is similar to performing a random walk in twodimensions and the total amplitude (distance travelled)is proportional to the number of steps, resulting in thenet interference contrast being proportional to 1 / √ L .Finally, for intermediate lengths and temperatures, weobserve the formation of a double-peak structure in thedistribution functions, characterized by a peak at zero,originating from thermal noise and a peak at finite am-plitude α originating from quantum noise [30, 40]. Inthis crossover regime the relative effects of quantum fluc-tuations are diminished but not completely suppressed,so that both quantum and thermal fluctuations are ofimportance in determining the shape of the distributionfunctions. We find very good agreement between exper-iment and theory also in this crossover regime. VI. DISCUSSION AND SUMMARY
It is interesting to note that the above analysis is basedon the Luttinger liquid theory of interacting bosons inone dimension which is accurate for calculating the long-range part of the correlation functions but does not cap-ture the short distance part on the scale of the healinglength ξ h . For system size L ≫ ξ h it is this long distancepart of the correlation functions that gives the dominantcontribution to the integrals determining the interferenceamplitude (see equation (2)). This sensitivity to the longdistance part of the correlation functions is the uniquefeature of interference experiments, which makes them apowerful tool for analyzing quantum and thermal fluctu-ations in low-dimensional condensates.Alternative approaches, such as measurements of den-sity fluctuations in expanding condensates [41] are prob-ing correlation functions on the scale of the healinglength. This short range part of the correlation func-tion is hardly sensitive to the quasi long-range nature ofquantum fluctuations in one-dimensional systems. Thismakes it difficult to observe quantum effects by directmeasurement of density fluctuations: they reveal the roleof interactions [42] but the transformation of short rangecorrelations into density fluctuations masks the quantumcorrelations.In summary, we have studied quantum and thermalnoise in 1d systems of interacting quantum degeneratebosons using the full distribution function of the inter-ference amplitude. The shot-to-shot fluctuations in thecontrast contain information, which can be related tohigh-order correlation functions of the 1d system. Ourresults provide the first experimental measurements ofthe full distribution function of quantum noise in an in-teracting many-body system. By analyzing these distri-bution functions we provide the first direct experimen-tal evidence of quasi long-range order in one-dimensionalcondensates. The remarkable agreement between our ex-perimental findings and theoretical predictions based onthe Luttinger liquid model provides a confirmation of thistheoretical approach as an effective low-energy theory ofinteracting bosons in one dimension. This demonstratesthe power of quantum noise analysis in studying stronglycorrelated many-body systems.We expect our experiments to pave the way for othermethods of characterizing many-body systems using theanalysis of quantum noise like particle number fluctua-tions [43] and spin noise [44, 45, 46]. From the point ofview of analyzing systems with strong interactions andcorrelations, this should allow cold-atom experiments toprovide a complementary and different perspective tothat provided by electron systems. VII. METHODSA. Preparing two independent 1d condensates onan atom chip
We start the experiment with a thermal ensemble of ∼ Rb atoms in the | F = 2 , m F = 2 > state at atemperature T ≈ µ K in a single highly elongated mag-netic trap on an atom chip [26, 27]. This initial sample isprepared using our standard procedure of laser cooling,magnetic trapping, and evaporative cooling [47]. Theinitial trapping configuration is then deformed along thetransverse direction into a highly anisotropic double-wellpotential by means of radio-frequency-induced adiabaticpotentials [23, 25]. In particular, we employ the rf-trapsetup presented in ref. [24] where the combination of tworadio-frequency (rf) fields generated by wires on the atomchip allows the realization of a compensated symmetricdouble-well potential in the vertical plane. The final cool-ing of the two separated ensembles leading to the two 1dcondensates is achieved by performing forced evaporativecooling in the dressed state potential [48]. We observe theonset of quantum degeneracy at T ≈
400 nK in each ofthe two potential tubes.The potential barrier between the two systems is con-trolled by the amplitude of the rf fields and the gradientof the static magnetic trap [25]. We realize a barrierheight V ≈ k B × µ K to ensure a complete decouplingof the two systems during the final cooling stage.After cooling and a relaxation time of 300 ms to ensureeach system is in equilibrium (a constant rf knife is kepton during this time to prevent heating), each potentialtube contains 3000 − T <
100 nK. The atoms are trapped in a strong trans-verse harmonic confinement of ν ⊥ ∼ . µ m from the atom chip surface. Each individ-ual degenerate atomic ensemble is in the one-dimensionalregime, with both temperature T and chemical potential µ fulfilling k B T, µ < hν ⊥ [28, 29]. B. Measuring the interference pattern
We observe the interference pattern created by the twoexpanding, overlapping atomic clouds using standard ab-sorption imaging. For the vertical double-well orienta-tion used in the experiments, the observed interferencefringes in the atomic density are horizontal, parallel tothe atom chip surface. This enables us to image the in-terference pattern along the transverse direction of thesystem. The employed imaging system has a spatial res-olution of 3 . µm and a noise floor of ∼ µm pixel. C. Extracting the fringe amplitude frominterference patterns
From a single interference image we obtain line pro-files for different L by integrating the two-dimensionalabsorption image over various lengths along the longitu-dinal direction of the system. The obtained line densi-ties are then Fourier transformed, and we extract A Q asthe value of this Fourier transform at the wave vector Q corresponding to the observed fringe spacing. This spac-ing is determined from fitting the interference patternswith a cosine function with a Gaussian envelope plus anunmodulated Gaussian to account for the contrast re-duction. The free parameters of these fits are the rela-tive phase θ , the contrast, and the fringe spacing. Thewidth, amplitude, and center position of the total cloudare determined independently from a Gaussian fit to thefull integrated density pattern of the central area of eachimage. Note that the absolute value of the interferenceamplitude | A Q | (as defined in eq. 1) and the contrast C are related as | A Q | = n d L × C . D. Average interference amplitude fits
To compare the experimentally observed length depen-dence of the average interference amplitude to theory, weperform a least square fit of the theoretical prediction(equation (3)) to the data. Since the line density n d is extracted directly from the absorption images and thetransverse trapping frequency ν ⊥ is measured preciselyby parametric heating experiments, the only unknownexperimental parameter is the temperature T . We in-clude a global additive fit parameter to account for con-trast reduction due to technical aspects such as finiteimaging resolution and focal depth. E. Luttinger liquid
A one-dimensional gas of ultra-cold bosonic atoms canbe described by the Lieb-Liniger model of bosons inter-acting via a point-like repulsion [21, 49, 50]. The ef-fective approach to the Lieb-Liniger model, capturingthe long-distance behavior of all correlation functions isknown as the Luttinger liquid formalism [35, 36, 37].The essence of this approach is to represent the origi-nal bosonic field in terms of the two phase fields a ( z ) =( n d + ∂θ ( z )) e iφ ( z ) and keep only the terms quadraticin φ ( z ) , θ ( z ) in the Hamiltonian. The resulting theoryhas linear spectrum of bosonic sound waves and showsalgebraic decay of all correlation functions at zero tem-perature (e.g < a † a > ∼ | z − z | / K ) and exponentialdecay for finite temperature (e.g. < a † ( z ) a ( z ) > ∼ n d [ π/ ( ξ T n d sinh( π ( z − z ) /ξ T )] / K . Here K is thefundamental parameter of the theory, the so-called Lut-tinger parameter and ξ T = ξ φ ( T ) /K . The last expressionapplies down to the short distance cut-off given by thehealing length. The value of K is uniquely determined bythe dimensionless ratio characterizing the original micro-scopic model: γ = mg/ ~ n d , where g is one-dimensionalinteraction strength. In the weakly interacting regimestudied here, K ≈ π/ √ γ . Recent analysis showed [51, 52]that the Luttinger liquid formalism provides extremelyaccurate description of the correlation functions of theLieb-Linger model for both long distances and distancesjust beyond the healing length. F. Calculation of the distribution functions
Computation of the distribution functions requires, inprinciple, the knowledge of all moments of the interfer-ence fringes amplitude. One approach to overcome thisproblem of moments was introduced in Ref. [19], wheremethods of conformal field theory and special proper-ties of exactly-solvable models were used to compute thedistribution function for periodic boundary conditions atzero temperature. Another method, which allows to com-pute the distribution functions for all boundary condi-tions, arbitrary temperature, and in all dimensions [40]is based on the mapping of the problem to a generalizedCoulomb gas model and a related problem of fluctuatingrandom surfaces (for a review, see Ref. [30]). This is theapproach which we use in our analysis.The full distribution function W ( α ) is defined by the normalized moments of the interference fringe contrastas h α m i = h| A Q | m i / h| A Q | i m = Z ∞ W ( α ) α m dα. 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