Extension of the Generalized Hydrodynamics to the Dimensional Crossover Regime
Frederik Møller, Chen Li, Igor Mazets, Hans-Peter Stimming, Tianwei Zhou, Zijie Zhu, Xuzong Chen, Jörg Schmiedmayer
EExtension of the Generalized Hydrodynamics to Dimensional Crossover Regime
Frederik Møller , Chen Li , Igor Mazets , Hans-Peter Stimming , Tianwei Zhou ,Zijie Zhu , Xuzong Chen , and J¨org Schmiedmayer Vienna Center for Quantum Science and Technology (VCQ), Atominstitut, TU Wien, Vienna, Austria School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China Wolfgang Pauli Institut, c/o Fakult¨at f¨ur Mathematik, Universit¨at Wien, 1090 Vienna, Austria Fakult¨at f¨ur Mathematik, Universit¨at Wien, 1090 Vienna, Austria INO-CNR Istituto Nazionale di Ottica del CNR,Sezione di Sesto Fiorentino, I-50019 Sesto Fiorentino, Italy Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland (Dated: June 26, 2020)In an effort to address integrability-breaking in cold gas experiments, we extend the integrablehydrodynamics of the Lieb-Liniger model with a second component. The additional componentrepresents the population of transversely excited atoms found in quasi-1d condensates. Collisionsbetween different components are accounted for through the inclusion of a Boltzmann-type collisionintegral in the hydrodynamic equation. We compare predictions of our model to measurements froma quantum Newton’s cradle setup at short to intermediate time scales and observe good agreement.
Over the last decades, the advances in experimentallyrealizing and manipulating quantum many-body systemsin low dimensions have increased the demand for theoret-ical methods capable of describing their complex dynam-ics [1–6]. Arguably one of the most prominent exper-imental platforms for studying out-of-equilibrium phe-nomena is ultracold Bose gases [7–21], which upon con-finement to one dimension exhibits integrability. Inte-grable systems abide to an extended set of conservationlaws, strongly constraining their dynamics and inhibit-ing thermalization [22–25]. Within the integrable limit,the recent theory of Generalized Hydrodynamics (GHD)has established itself as a powerful and flexible frame-work by capturing both the transport of all the conservedcharges and the Wigner delay time in elastic scatteringof particles [26] within a single continuity equation [27–29]. For earlier applications of the Wigner delay time tothe hydrodynamics of one-dimensional, non-degenerategases see Refs. [30, 31]. Building upon the framework ofGHD, a wide array of extensions have enabled the studyof correlations [32–35], Drude weights [36–39], diffusionconstants [40–43], and more.However, real systems realized in even very controlledenvironments are only approximately integrable. Smallexperimental imperfections and processes outside therealm of GHD will break the integrability of the systemand eventually over time drive it towards thermalization.Recently, the topic of thermalization has attracted a lotof attention [44–48], however, a generally applicable the-ory appears intractable as mechanisms of integrablity-breaking often depend on the physical realization of thegiven system [49–57]. Hence, for now, thermalizationmust be accounted for through considerations of the ex-perimental circumstances [58].In this Letter, we seek to extend the applicability ofGHD to the dimensional crossover regime, which is ac-cessed when the collisional energy of atoms exceeds the
FIG. 1: Mechanism for thermalization in quasi-1d Bose gas.Two atoms in the transverse ground state collide with largeopposite momenta, exciting one of the atoms to the secondexcited state. The excited atom can decay to the ground statethrough collisions with ground state atoms. level spacing of the transverse confinement. Thus, basedon heuristic considerations, we introduce a second com-ponent to the Lieb-Liniger model, representing atomsin the second transversely excited state. The couplingbetween components is accounted for by introducing aBoltzmann-type collision integral to the GHD equation.We then compare the predictions of our model to ex-perimental results from a quantum Newton’s cradle-typesetup. To demonstrate the effects of the second compo-nent, we perform the same calculations using standardGHD and quantitatively compare the two approaches.The degenerate gas of N bosonic atoms of mass m isdescribed by the second-quantized Hamiltonianˆ H = (cid:90) d r (cid:110) (cid:126) m ( ∇ ˆΨ † )( ∇ ˆΨ) + [ U ( z ) + V ⊥ ( x, y )] ˆΨ † ˆΨ+2 π (cid:126) a s m ˆΨ † ˆΨ † ˆΨ ˆΨ (cid:111) , (1)where ˆΨ = ˆΨ( r ) is the atom annihilation operator, a s is the s -wave scattering length, U ( z ) is the loose trap-ping potential in the longitudinal direction, V ⊥ ( x, y ) isthe tight transverse trapping potential. We assume that V ⊥ ( x, y ) is harmonic and axially symmetric, ω ⊥ being its a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un fundamental frequency and l ⊥ = (cid:112) (cid:126) / ( mω ⊥ ) being thecorresponding length scale.We treat the motion of atoms in the longitudinal di-rection within the GHD framework, while the transversemotion is accounted for via a collisional integral. TheGHD provides a coarse grained theory for the dynam-ics of systems close to an integrability point [27, 28].Just like the thermodynamics Bethe ansatz, the the-ory encodes the thermodynamic properties of a localequilibrium macrostate in a distribution of quasiparti-cles [59, 60]. Each quasiparticle is uniquely labelled by itsrapidity, θ . Following the original paper by Lieb and Lin-iger [61, 62], we express rapidity in inverse length units.In the thermodynamics limit, the rapidity becomes a con-tinuous variable, with the density of occupied rapiditiesin the phase ( z, θ )-space given by the time-dependentquasiparticle density, ρ p ( z, θ, t ). Similarly, one can in-troduce a density of holes, ρ h ( z, θ, t ), describing the den-sity of unoccupied rapidities [63]. Together these twodensities describe the density of states and obey the rela-tion ρ p ( θ ) + ρ h ( θ ) = (2 π ) − + π − (cid:82) ∞−∞ dθ (cid:48) { c/ [ c + ( θ (cid:48) − θ ) ] } ρ p ( θ (cid:48) ), where c = 2 a s /l ⊥ is the interaction param-eter of the Lieb–Liniger model (we assume a s (cid:28) l ⊥ ).Here we omit the co-ordinate and time arguments whenappearing the same in all terms. A quasiparticle withrapidity θ propagates at velocity v eff , which obeys theintegral equation v eff ( θ ) = (cid:126) θ/m + (cid:82) ∞−∞ dθ (cid:48) { c/ [ c +( θ (cid:48) − θ ) ] } ρ p ( θ (cid:48) )[ v eff ( θ (cid:48) ) − v eff ( θ )] and encodes the Wigner de-lay time associated with the phase shifts occurring underelastic collisions in integrable systems [37, 64]. In anexternal potential, a force F eff = − ∂ z U ( z ) acts on thequasiparticles.If two atoms with rapidities θ and θ (cid:48) collide and thecollision energy exceeds 2 (cid:126) ω ⊥ , their transverse states canchange, as illustrated in figure 1. Two collision outcomesare equally probable: (i) one atom remains in the trans-verse ground state and the other one occupies the secondexcited state; (ii) both of the atoms are transferred tothe first excited state. Parity conservation plays an im-portant role here. First of all, transitions of only oneatom to the first transversely excited state are forbid-den, since this state is odd (has a negative parity), incontrast to the ground and second excited states, whichare even. Likewise, the de-excitation of an atom in thetransverse second excited state happens due to a collisionwith an atom in the ground state. This process occursat a higher rate than the de-excitation of an atom in thefirst excited state, since a collision with another excitedatom is a much more rare event in the regime where mostof the atoms are confined to the transverse ground state.In the presence of these processes, our extended modelyields ∂ t ρ p + ∂ z ( v eff ρ p ) + (cid:126) − ∂ θ ( F eff ρ p ) = I ( θ ) . (2)Eq. (2) differs from the conventional GHD equation by the Boltzmann-type collision integral I ( θ ) in the right-hand side.We take into account only collisional transitions be-tween the transverse ground state and the second excitedstate, since this process leads to fast relaxation of therapidity distribution. Slow relaxation via double popu-lation of the first excited state is neglected in order tomake the model computationally fast and efficient. Thisexclusion of one of the transverse excitation channels isaccounted for by reducing the probability of transverse-state changing collisions by a factor ζ ≈ .
5. Therefore,to the simplest approximation, the collision integral canbe written as I ( θ ) = −I − p ( θ ) + I − h ( θ ) ν − I + p ( θ ) ν + I + h ( θ ) , (3)where ν is the probability for an atom to be in the trans-versely excited state with the excitation energy 2 (cid:126) ω ⊥ .We assume ν (cid:28)
1. The terms in Eq. (3) are defined as I ± α ( θ ) = (2 π ) (cid:126) m (cid:90) R ± dθ (cid:48) | θ − θ (cid:48) | P (cid:108) ( | θ − θ (cid:48) | , | θ ± − θ (cid:48)± | ) × ρ α ( θ ) ρ α ( θ (cid:48) ) ρ ¯ α ( θ ± ) ρ ¯ α ( θ (cid:48)± ) , (4)where ¯ α = h for α = p and vice versa, P (cid:108) ( θ , θ ) =4 ζc θ θ / [ θ θ + c ( θ + θ ) ], θ ± = ( θ + θ (cid:48) ) + ( θ − θ (cid:48) ) (cid:112) ± / [( θ − θ (cid:48) ) l ⊥ ] , and θ (cid:48)± = ( θ + θ (cid:48) ) − ( θ − θ (cid:48) ) (cid:112) ± / [( θ − θ (cid:48) ) l ⊥ ] . The integration ranges in Eq(4) are following: R + is the whole real axis and R − iscomprised of those real values of θ (cid:48) , which yield real θ − and θ (cid:48)− , i.e., R − = { θ (cid:48) : θ (cid:48) < θ − √ /l ⊥ } ∪ { θ (cid:48) : θ (cid:48) >θ + 2 √ /l ⊥ } .If two atoms in different transverse states collide, thetransverse state exchange is a relatively highly proba-ble outcome. Therefore, we neglect correlations betweentransverse excitations and rapidities and introduce ν ( t ),which is uniform in the phase space and obeys a simpleequation dνdt = Γ + h − Γ + p ν + γ, (5)where Γ + α = (2 N ) − (cid:82) ∞−∞ dz (cid:82) ∞−∞ dθ I + α ( θ ), α = p, h , and γ accounts for any heating rate caused by experimentalimperfections.To demonstrate the applicability of the two-componentLieb-Liniger model, we use our method to provide a qual-itative description of the initial relaxation in a quantumNewton’s cradle setup and compare our calculations todata from an experiment which explores the onset of thedimensional crossover [65].The experiment (for a detailed description see [65])studies the dynamics of Rb Bose-Einstein condensatesin a 2d lattice of independent 1d traps with a tight trans-verse confinement of ω ⊥ / π = 31 kHz and weak longitu-dinal confinement of ω (cid:107) / π = 83 . Degen. Non-Degen.
FIG. 2: Evolution of Bose gas in the first 60 periods of the cradle. Top and middle row display the quasiparticle density of thestandard and our extended GHD, respectively. See text for details of the simulation. In the final row, the rapidity distributionsof the quasiparticles are compared to the bosonic MDFs measured in the experiment. Initially the two quantities are verydifferent, however, as the gas becomes increasingly non-degenerate, the two distributions become increasingly alike. τ = 12 ms). Owing to the Gaussian profile of the trap-ping beams (beam-waist of σ = 145 µ m), the longitudinalpotential is slightly anharmonic. The weighted-averageatom number per tube is between 80 and 120, providinga coupling strength in the intermediate regime.The dynamics in the longitudinal direction are initi-ated by two Bragg pulses, imparting opposite momentaof ± (cid:126) k Bragg to each half of the atomic cloud, with k Bragg = 2 π/
852 nm. The momentum kick correspondsto ∼
40% of the excitation energy (2 (cid:126) ω ⊥ ).After the Bragg pulses, the atoms oscillate and col-lide with each other under the 1d confinements for someduration. At the end of the evolution, the atoms arereleased from the optical lattice and detected after time-of-flight (TOF) providing us with the momentum distri-bution function in the longitudinal direction. Very lowheating rate, about 40 nK/s, and negligible atom loss,smaller than 5% within the concerned time scale in thisLetter, are observed in experiments.The initial state of the simulation is obtained in a man-ner similar to Ref. [57, 66]; we assume the pre-pulsequasiparticle density to be a thermal state and modelthe Bragg pulse sequence as shifts of the distributionalong the rapidity axis. In the experiment we observea small number of atoms leftover at low momenta afterthe pulse sequence. We account for those by leaving afraction η = 0 .
07 of the quasiparticle density un-shifted.Thus, our initial state reads ρ init p ( θ ) = (1 − η ) ρ th .p ( θ +2 k Bragg ) + (1 − η ) ρ th .p ( θ − k Bragg ) + ηρ th .p ( θ ).In the following, we study the case of N = 80 atomsper tube at a temperature of 80nK, leaving around 1.5%of the atoms above the excitation threshold. The same analysis was also performed for a data set with N = 120yielding similar results. We focus on the first 60 oscil-lation periods of the cradle (720ms), where interactionsof the atoms play a significant role and previous studieshave shown dephasing processes to dominate the dynam-ics [55, 65]. Thus, methods like GHD are required toproperly describe this regime. At later times the gas issufficiently deep in the non-degenerate regime, enablingthe use of molecular dynamics simulations. For discus-sions of the numerical methods used for the simulation,see Supplemental Material and Ref. [67].Figure 2 shows a side-by-side comparison of the quasi-particle density of the standard and two-component Lieb-Liniger model. While the Bragg peaks of the initial statepersist throughout the evolution when propagated usingthe standard GHD equation, the inclusion of the colli-sion integral enables quasiparticles to distribute acrossthe phase space. Hence, the additional component ap-pears to accelerate (or in this case enable) the dephasing.This is contrary to Ref. [57], where the anharmonic trap-ping potential was sufficient to induce dephasing. How-ever, the dephased state there was distinctly differentfrom thermal. Further, the initial population of atomsat low rapidities is depleted in the standard GHD, givingrise to ’self-stabilization’ of the Bragg peaks. Reducingthe interaction between atoms or increasing the degreeof anharmonicity in our simulation yields results simi-lar to Ref. [57]. Thus, we attribute our observationsto a certain combination of parameters, further exem-plifying the complex dynamics possible in the quantumNewton’s cradle. Nevertheless, tuning these parametershas only little influence on the final state obtained via the FIG. 3: Evolution of the variance of the MDF (or RDF)over a period, quantifying aspects of the dephasing and ther-malization. The inset shows the evolution of the excitationprobability in the extended model. The shaded area indicatesthe noise floor, where the variance is dominated by noise inthe experiment. two-component model, which apart from the now barelyvisible Bragg peaks strongly resembles a thermal state.In the final row of figure 2 we compare the rapidity dis-tribution (RDF) of the quasiparticles f ( θ ) = (cid:82) d z ρ p ( θ, z )with the measured momentum distribution (MDF) of theBose gas. It is important to note that the two quantitiesare inherently different; while evolving in the 1d cradlethe atoms follow the rapidity distribution, however, uponmeasurement the entire lattice is ramped down causingthe atoms to expand in three dimensions. After time-of-flight expansion in 3d, the recorded density profile ofthe atoms corresponds to the MDF [68]. The relationbetween the different distributions is not easily obtained.For short times we can estimate the MDF from the ra-pidities (see Supplemental Material or Ref. [4]), whilein the non-degenerate regime the two distributions coin-cide [69]. We observe this in figure 2, where the measuredMDF approaches the RDF over time. The main driverof this transition is the dephasing, which lowers the den-sity of atoms, thus bringing the gas towards the non-degenerate regime. After 60 oscillation periods (720ms),computations of the g -function for the extended modelyielded g ∼ .
79, thus indicating that the gas is stillnot completely non-degenerate. Nevertheless, we observegood agreement between the MDF and the RDF of thetwo-component model, although our model features moreprominent high-rapidity tails.To further quantify the differences between the GHDsimulations and the experiment, we plot in figure 3 thevariance of the RDF (and MDF) profiles over one period.The quantity D ( t ) = (cid:82) d θ (cid:82) τ d t (cid:48) [ f ( t + t (cid:48) , θ ) − F ( t, θ )] ,where F ( t, θ ) is the mean profile, provides insight in boththe dephasing and thermalization of the gas [65]. Fur- ther, this measure is fairly robust with regards to the dif-ference in observable, although we initially observe somediscrepancy between experiment and simulations. Never-theless, the observed dephasing of the experiment and thetwo-component model become quite similar after around20 periods (240ms) and actually start coinciding as theMDF and RDF become alike. Hence, qualitatively, thepredictions of the two-component model agree with theexperiment. At longer times, experimental noise startsdominating the calculation of D ( t ) causing it to plateau.We indicate this noise floor by the shaded area in the fig-ure. Meanwhile, the standard GHD maintains the Braggpeaks throughout the evolution causing no noticeable de-phasing.The inset of figure 3 shows the evolution of the excita-tion probability ν , which we assume to be zero initially.Importantly, the value of ν will always tend towards adynamic equilibrium, as any excess excitations will de-cay faster than new excitations are produced. Rather re-markably we observe that just a small fraction of excitedatoms can dramatically change the dynamics in the cra-dle. We emphasize that the influence of excited states isnot limited to Newton’s cradle setups; reducing the trans-verse trapping frequency or increasing the temperaturewill both lead to an increased probability of transverseexcitations [18, 70–73]. For instance, a thermalized Bosegas with temperature T = (cid:126) ω ⊥ / SUPPLEMENTARY MATERIALStandard GHD equations
We report here the equations for the standard GHDof the Lieb-Liniger model. Note, we omit all spacial andtemporal arguments, as they will remain the same oneither side of the equations.The standard GHD propagation equation reads ∂ l ρ p + ∂ z (cid:0) v eff ρ p (cid:1) + (cid:126) − ∂ θ (cid:0) F eff ρ p (cid:1) = 0 , (6)where the effective force on the quasiparticles F eff de-scribes changes in the rapidity distribution in the pres-ence of inhomogeneous interactions, while the effectivevelocity is given by v eff ( θ ) = (cid:126) θm + (cid:90) ∞−∞ dθ (cid:48) Φ( θ, θ (cid:48) ) ρ p ( θ (cid:48) ) (cid:2) v eff ( θ (cid:48) ) − v eff ( θ ) (cid:3) . (7)Here, Φ( θ, θ (cid:48) ) = cc +( θ − θ (cid:48) ) is the Lieb-Liniger two-bodyscattering kernel. From the quasiparticle density, one canextract the expectation values of the conserved chargesand their associated current, respectively, viaq i = (cid:90) d θ h i ( λ ) ρ ( λ ) (8)j i = (cid:90) d θ h i ( λ ) v eff ( λ ) ρ ( λ ) , (9)with h i ( λ ) being the one-particle eigenvalue of the i ’thconserved charge.As an alternative to the quasiparticle density, one canencode the thermodynamic properties of the system inthe filling function ϑ ( θ ) = ρ p ( θ ) ρ p ( θ ) + ρ h ( θ ) , (10)where the density of states is given by ρ p ( θ ) + ρ h ( θ ) = 12 π + 1 π (cid:90) ∞−∞ d θ (cid:48) cc + ( θ − θ (cid:48) ) ρ p ( θ (cid:48) ) . (11)The quasiparicles of the Lieb-Liniger model followFermionic statistics. Thus, a thermal state can be calcu-lated from ϑ ( θ ) = 11 + e (cid:15) ( θ ) β , (12) where β is the inverse temperature and the pseudoenergy (cid:15) ( θ ) is acquired from solving the equation (cid:15) ( θ ) = (cid:126) θ m − µ − πβ (cid:90) ∞∞ d θ (cid:48) cc + ( θ − θ (cid:48) ) ln (cid:16) e (cid:15) ( θ (cid:48) ) β (cid:17) . (13)The chemical potential µ ( z ) = µ − U ( z ) accounts for theexternal potential. Numerically solving the propagation equation
For the numerical GHD computations we employ theiFluid library [67]. In order to solve Eq. (2) we employa first order split step propagation scheme.First, we evaluate the collision integral, which requiresquantities readily available from GHD. Throughout theentire calculation we maintain the same rapidity and col-lision grids. Consider the rapidity discretized on a grid θ i with i = 1 , . . . , i max . The collision grids then read θ ± [ i ; j ] = 12 ( θ i + θ j )+sgn ( θ i − θ j ) (cid:113) ± / [( θ i − θ j ) l ⊥ ] (14)where θ (cid:48)± [ i ; j ] = θ ± [ j ; i ]. To obtain the particle and holedensities on the collision grids we use interpolation, whichcan be expressed in matrix form as ρ p,h ( θ ± [ i ; j ]) = (cid:88) k Ξ ± ([ i ; j ] , k ) ρ p,h ( θ k ) . (15)Throughout the simulation we maintain constant rapid-ity and collision grids. Thus, the interpolation matrixΞ ± can be calculated beforehand, greatly reducing thecomputational time needed. For linear interpolation, theinterpolation matrix can be constructed as followsΞ ± ([ i ; j ] , k −
1) = θ k − θ ± [ i ; j ] θ k − θ k − (16)Ξ ± ([ i ; j ] , k ) = 1 − θ k − θ ± [ i ; j ] θ k − θ k − , (17)where k is the index minimizing min k | θ k − θ ± [ i ; j ] | . Thismatrix structure is sparse, allowing for very fast interpo-lation.Once the collision integral has been obtained, we solvethe equationsdd t ρ p ( θ, z, t ) = I ( θ, z, t ) (18)dd t ν ( t ) = Γ + h ( t ) − Γ + p ( t ) ν ( t ) + γ (19)using the two-step AdamsBashforth method. Next, wesolve the standard GHD equation (6) without collisionintegral using the method of characteristics. Here weemploy the second order scheme detailed in Ref. [29]. Details of the simulation
The quantum Newtons cradle is realized experimen-tally in a red detuned optical lattice consisting of many1D tubes. By preparing a specific number of atoms be-fore the lattice is ramped up we can adjust the numberof atoms per tube. Nevertheless, the atom number willnot be the same for all tubes. Particularly between innerand outer tubes will differences occur. Upon measure-ment, the contribution from all tubes are averaged.In the GHD simulation we treat only a single tube, us-ing parameters obtained from the weighted average overall the tubes. Thus, we study the case of a tube with N = 80 atoms at an initial of temperature 80nK. Thelongitudinal potential emerges naturally from the Gaus-sian intensity profile of the lattice beams U ( z ) = mω (cid:107) σ (cid:16) − e − z /σ (cid:17) , (20)where ω (cid:107) / π = 83 . σ = 145 µ m is the beam-waist of the latticebeams. We note that the σ of a given tube will be slightlylarger, if the optical lattice is not perfectly aligned.The heating process is studied by observing the evo-lution of a cloud held in the optical lattice without theBragg-pulse excitation. Over time, heating effects fromthe trapping laser will cause the momentum peak of thecloud to expand over time. From the MDF we can com-pute the kinetic energy of the gas, and any increase inkinetic energy is attributed heating effects. For N = 80atoms, we estimate a heating rate of 40nK/s. We do nottake into account atom losses in the simulation.We performed several simulations with slightly per-turbed parameters and observed no significant change inthe outcome. Setting up the two-component model
To construct a numerically tractable extension of theGHD, we need to assume several simplifications and ap-proximation. One possible path towards thermalizationis through collisions with transversely excited atoms.Parity conserving collisions of atoms in the transverseground state with sufficiently high collision energy canlead to excitation of either one atom into the secondtransversely excited state or two atoms into the first ex-cited state. We will neglect this distinction and assumethat the system contains only two components: atoms inthe transverse ground state (denoted as the pseudospinstate | ↓(cid:105) ) and atoms in the axially symmetric trans-verse state with the excitation energy 2 (cid:126) ω ⊥ (denoted by | ↑(cid:105) ). The Bethe-ansatz solution for an integrable two-component 1D Bose gas was first proposed by Yang [83].Eigenstates of the two-component 1D Bose gas are char- acterized not only by the quasiparticle rapidities, but alsoby rapidities λ of pseudospin waves.The bosonic wave function of N bosons is symmetricwith respect to permutations of atoms. For an eigenstate,it can be writted as an irreducible tensor product of thepseudospin and co-ordinate parts, each of them belongingto the same irreducible representation of the symmetricgroup S N . An irreducible representation of S N is denotedby the corresponding Young diagram. Since only twopseudospin states are present, the Young diagram cancontain maximally 2 rows, i.e., has the form { N − M, M } where M is an integer from 0 to the integer part of N/ M is not the number of atoms in the state | ↑(cid:105) ;the latter number is larger than or equal to M .In the general case, pseudospin rapidities can be com-plex, forming so-called Bethe strings. However, since thefraction of atoms in the | ↑(cid:105) state is small, we can assumeIm λ = 0. Thus, we can introduce quasiparticle (p) andhole (h) distributions σ p,h ( λ ) for the pseudospin rapidi-ties as well. Because M (cid:28) N , the contribution of thepseudospin component to the quasimomenta density ofstates is negligible. Therefore, we can roughly estimate σ p ( λ ) + σ h ( λ ) ≈ ρ p | θ = λ . (21)Eq. 21 has a clear physical meaning: each atom canbear, additionally to its quasimomentum, a pseudospinexcitation. Thus, we denote the probability of an atombearing a pseudospin excitation by ν ( θ ) = σ p σ p + σ h (cid:12)(cid:12)(cid:12)(cid:12) λ = θ ≈ σ p ρ p (cid:12)(cid:12)(cid:12)(cid:12) λ = θ (22)and assume in the following that ν (cid:28) ∂ t ρ p + ∂ x ( v eff ρ p ) + (cid:126) − ∂ θ ( F eff ρ p ) = I ( θ ) . (23)In the following, we will derive an expression for the col-lision integral. Collision integral
We consider atoms in a waveguide under assumptionthat the collision energy may exceed 2 (cid:126) ω ⊥ , but is cer-tainly below 4 (cid:126) ω ⊥ . We extend Olshanii’s treatment [74]to collision energies high enough to excite the transversedegrees of freedom. The renormalized coupling strengthis then ˜ c = c − cl ⊥ C ( (cid:15) ) , (24)where C ( (cid:15) ) ≈ (cid:113) − (cid:15) − (cid:113) − (cid:15) − √ √ − (cid:15) , (25) (cid:15) = ( k − k ) l ⊥ , and (cid:126) k , (cid:126) k are the mo-menta of colliding bosonic atoms. Eq. (25) is de-rived using the simplest approximation to the sum (cid:80) ∞ n =2 exp( −√ √ n − (cid:15) | z | /l ⊥ ) √ √ n − (cid:15) ≈ (cid:82) ∞ dn (cid:48) exp( −√ √ n (cid:48) − (cid:15) | z | /l ⊥ ) √ √ n (cid:48) − (cid:15) +
12 exp( −√ √ − (cid:15) | z | /l ⊥ ) √ √ − (cid:15) according to the Euler–Maclaurinformula. This approximation is quite good, since Eq.(25) yields C (0) ≈ .
04, while the exact result is C (0) =1 . . . . .For cl ⊥ (cid:28) c is close to c for almostall collision energies, except of a narrow interval near theexcitation threshold (cid:15) = 1. If (cid:15) >
1, the imaginary partof ˜ c is non-zero, which corresponds to the probability ofexcitation of the transverse degrees of freedom P ( k, q ) = 4 c kqk q + c ( k + q ) , (26)where k = | k − k | , q = (cid:113) | k − k | − l − ⊥ , (27)We will use the dimensional coupling constant c and theexcitation probability (27) as basic building blocks forour extended GHD. The effective three-body elastic scat-tering of atoms via virtual excitation of the transversedegrees of freedom will be fully neglected in our theory.A collision that leads to excitation of transverse de-grees of freedom with the energy transfer 2 (cid:126) ω ⊥ has two equally probable outcomes: (i) one atom remains in thetransverse ground state and the other atom occupies thesecond excited state, (ii) both the atoms occupy the firstexcited state. In the former case, the excited atom can bede-excited by a collision with an atom in the ground state.In the latter case, de-excitation requires a collision of twotransversely excited atoms, which is much less probable,since, by assumption, the population of transversely ex-cited states is small. Therefore, the channel (ii) leads tomuch slower relaxation (change of the rapidity distribu-tion) than the channel (i). To simplify our equations andto make their numerical implementation more efficient,we neglect the channel (ii) and, respectively, reduce theprobability of the transverse-state changing collision bya factor ζ ≈ . P (cid:108) ( k, q ) = ζ P ( k, q ) . (28)We also introduce the quasimomenta of the atoms aftera collisional excitation or de-excitation of the transversestate as θ ± = ( θ + θ (cid:48) ) + ( θ − θ (cid:48) ) (cid:112) ± / [( θ − θ (cid:48) ) l ⊥ ] ,and θ (cid:48)± = ( θ + θ (cid:48) ) − ( θ − θ (cid:48) ) (cid:112) ± / [( θ − θ (cid:48) ) l ⊥ ] . Themicroscopic collision velocity is (cid:126) | θ − θ (cid:48) | /m . Knowing thescattering probability P (cid:108) , we can write the Boltzmann-type collision integral as I ( θ ) = (2 π ) (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ − − θ (cid:48)− | ) | θ − θ (cid:48) | Θ( | θ − θ (cid:48) | l ⊥ − √ (cid:110) − ρ p ( θ ) ρ p ( θ (cid:48) ) ρ h ( θ − ) ρ h ( θ (cid:48)− )+12 ρ h ( θ ) ρ h ( θ (cid:48) ) ρ p ( θ − ) ρ p ( θ (cid:48)− )[ ν ( θ − ) + ν ( θ (cid:48)− )] (cid:111) +(2 π ) (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ + − θ (cid:48) + | ) | θ − θ (cid:48) | (cid:110) − ρ p ( θ ) ρ p ( θ (cid:48) ) ρ h ( θ + ) ρ h ( θ (cid:48) + )[ ν ( θ ) + ν ( θ (cid:48) )]+ ρ h ( θ ) ρ h ( θ (cid:48) ) ρ p ( θ + ) ρ p ( θ (cid:48) + ) (cid:111) , where Θ( x ) is the Heaviside step function. For ν that does not depend on θ we obtain I ( θ ) = (2 π ) (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ − − θ (cid:48)− | ) | θ − θ (cid:48) | Θ( | θ − θ (cid:48) | l ⊥ − √ (cid:104) − ρ p ( θ ) ρ p ( θ (cid:48) ) ρ h ( θ − ) ρ h ( θ (cid:48)− )+ ρ h ( θ ) ρ h ( θ (cid:48) ) ρ p ( θ − ) ρ p ( θ (cid:48)− ) ν (cid:105) +(2 π ) (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ + − θ (cid:48) + | ) | θ − θ (cid:48) | (cid:104) − ρ p ( θ ) ρ p ( θ (cid:48) ) ρ h ( θ + ) ρ h ( θ (cid:48) + ) ν + ρ h ( θ ) ρ h ( θ (cid:48) ) ρ p ( θ + ) ρ p ( θ (cid:48) + ) (cid:105) . The key idea behind the expression for the collision inte-gral is that in the quantum degenerate regime the scat- tering is affected by the Pauli blocking: scattered atomscan acquire only those values of quasimomentum, whichwere not occupied before the collision. Therefore, thethe collision integrals must contain not only particle dis-tribution functions, but also hole distribution functions.Here the fermionic nature of particles and holes in theLieb-Liniger model is manifested. Note, that ρ h ( θ ± ) isthe Pauli blocking factor (1 minus the population) timesthe density of states for the scattering products. Fac-tor (2 π ) arises from the normalization. One factor 2 π arises from (cid:82) dt exp[ − i ( E i − E f ) t/ (cid:126) = 2 π (cid:126) δ ( E i − E f ),where E i , E f are the energies of the initial and finalstates, respectively. Another factor 2 π appears when weswitch from summation over discrete rapidities definedby the periodic boundary conditions over the length L to the integration over continuous θ ± : the Kroneckerdelta-symbol for discretized total momentum, δ P i ,P f =sinc [( θ ± + θ (cid:48)± − θ − θ (cid:48) ) L/ x = sin x/x , trans-forms to a 2 πδ ( θ ± + θ (cid:48)± − θ − θ (cid:48) ) /L , when we replacethe discrete sum by L (cid:82) dθ ± . . . , recall the normalization L (cid:82) dθ ρ p ( θ ) = N .The kinetics of pseudospin waves is more complicated.However, we can expect that the pseudospin transfer oc-curs on time scales much shorter than excitation of trans-verse modes. Therefore, we can assume that ν does notdepend on θ . As a further simplification, we assume that ν is also spatially uniform and depends on t only. In thisapproximation, dν ( t ) dt = (2 π ) (cid:126) mN (cid:90) ∞−∞ dz (cid:90) ∞−∞ dθ (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ + − θ (cid:48) + | ) | θ − θ (cid:48) | ρ p ( θ + ) ρ p ( θ (cid:48) + ) ρ h ( θ ) ρ h ( θ (cid:48) ) − (2 π ) (cid:126) mN (cid:90) ∞−∞ dz (cid:90) ∞−∞ dθ (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ + − θ (cid:48) + | ) | θ − θ (cid:48) | ρ h ( θ + ) ρ h ( θ (cid:48) + ) ρ p ( θ ) ρ p ( θ (cid:48) ) ν ( t ) . The collision integral I ( θ ) is identically zero when ra-pidities obey the Fermi–Dirac distribution and the classi-cal (Boltzmann) statistics holds for transverse excitations(recall that ν (cid:28) (cid:126) ω ⊥ /k B . For a non-degenerate 1D Bose gas, ρ p ( θ ) (cid:28) ρ h ( θ ) ≈ / (2 π ), thecollision integral takes the limit (Boltzmann) limit I cl ( θ ) = (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ − − θ (cid:48)− | ) | θ − θ (cid:48) |× Θ( | θ − θ (cid:48) | l ⊥ − √ × (cid:104) − ρ p ( θ ) ρ p ( θ (cid:48) ) + ρ p ( θ − ) ρ p ( θ (cid:48)− ) ν (cid:105) + (cid:126) m (cid:90) ∞−∞ dθ (cid:48) P (cid:108) ( | θ − θ (cid:48) | , | θ + − θ (cid:48) + | ) | θ − θ (cid:48) |× (cid:104) − ρ p ( θ ) ρ p ( θ (cid:48) ) ν + ρ p ( θ + ) ρ p ( θ (cid:48) + ) (cid:105) . Estimation of the bosonic MDF
The calculation of the bosonic MDF in Lieb-Linigermodel is not accessible with any general analytic method,and has only been done in numerical works [57, 68, 75–80]. Within the scope of this Letter, a rough estimationof MDF is sufficient to help us on demonstrating the ap-plicability of our model and comparing the results withexperimental measurements. Supposing that we have a density distribution of quasi-particles ρ p ( z, θ ) obtained from GHD. We regard it as atarget distribution and fit it with a sum of three thermaldistributions ρ ( z, θ ) = (cid:80) ρ j ( z, θ ) ( j = 1 , , ρ )is centered at the origin of phase space, and the other two( ρ and ρ ) are shifted by the mean rapidity boosts (cid:104) θ j (cid:105) .For each of these distributions which occur to be quiteclose to boosted thermal ones, we find chemical potentials µ j and temperatures T j , so that we can estimate theMDFs w j ( k ) with the following equations.In the degenerate limit, the MDF for Luttinger liquid,which is the Fourier transform of the correlation function,is expressed via Euler beta-function [81] w ( k ) = C K π k T Re[ B ( ik πk T + 14 K , − K )] , (29)where B ( x, y ) = Γ( x )Γ( y ) / Γ( x + y ), C ∼ c s is thespeed of sound, k T = k B T / ( (cid:126) c s ) and K = π (cid:126) n d / ( mc s )is the Luttinger liquid parameter. This profile includesa Lorentzian-shape peak and a pedestal decrease ∝ k − for k (cid:29) k T . For much larger momenta, the MDF is de-termined by Tan’s contact. There are known approachesto a precise calculation of the value of Tan’s contact us-ing, see, e.g., Ref. [82]. However, due to experimentaluncertainties, we use as a reasonable approximation thefolllowing modification of the MDF:˜ w ( k ) = w ( k ) (cid:113) ( kξ h ) [1 + ( kξ h ) + kξ h (cid:113) ( kξ h ) ] . (30) FIG. 4: Estimation of the mean momentum distributionfunction (MDF) over the first oscillation period of the cradlewith N = 120 atoms. The estimation is obtained from therapidity distribution (RDF) and is compared with the experi-mentally measured profile. On the x-axis θ is the rapidity forthe RDF, while we let θ = k for the MDF in order to comparethe two. When k strongly exceeds the inverse healing length ξ − h ,˜ w ( k ) ∝ k − .In the non-degenerate limit, the MDF coincides withthe quasimomentum distribution and approaches theMaxwell-Boltzman distribution at temperature T .On the basis of our experimental condition, the best-fit thermal distribution ρ is close to the non-degeneratelimit, so we take w ( k ) = (cid:82) d z ρ ( z, k ). While in theearly stage of evolution, ρ and ρ is deep in the degen-erate regime, and the MDFs w ( k ) and w ( k ) are writtenfollowing Eq. (29),(30). Since the mean quasimomentumequals to the mean momentum, these peaks are shiftedto be centered at k j = (cid:104) θ j (cid:105) . The three MDFs are normal-ized to the respective particle numbers N j , subject to therestriction (cid:80) N j = N . And the MDF w ( k ) = (cid:80) w j ( k ).In practice, discerning which atoms correspond towhich peak can be difficult, especially at the onset ofdephasing. Thus, we can only use the MDF estima-tion to check whether our initial state matches experi-mental observations. Additionally, we can estimate theperiod-mean profile by fitting the radial rapidity distri-bution in the ( x, θ ) phase-space. In theory this enablesdirect comparison between GHD and experiment, how-ever, measurements were performed at time intervals of1ms, yielding 12 different profiles per period. This resultsin a high degree of symmetry, effectively only probing 4different positions of the peaks (before dephasing). Thus,at short time-scales where the MDF estimation is valid,the measured mean MDF does not reflect the true mean.However, for N = 120 atoms we do have measurements ofthe very first period taken with a much finer time resolu- tion (measurement every 0.2ms). Hence, we can compareour estimated MDF directly to the experiment, as seen infigure 4. 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