diff --git "a/09FAT4oBgHgl3EQfChxb/content/tmp_files/2301.08410v1.pdf.txt" "b/09FAT4oBgHgl3EQfChxb/content/tmp_files/2301.08410v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/09FAT4oBgHgl3EQfChxb/content/tmp_files/2301.08410v1.pdf.txt" @@ -0,0 +1,3907 @@ +Caustics in the sine-Gordon model from quenches in coupled 1D Bose gases +Aman Agarwal,1, 2, 3, 4, 5, 6, ∗ Manas Kulkarni,3, † and D. H. J. O’Dell1, ‡ +1Department of Physics and Astronomy, McMaster University, +1280 Main St. +W., Hamilton, Ontario, Canada L8S 4M1 +2BITS-Pilani, K. K. Birla Goa Campus, NH17B, Bypass Road, Zuarinagar, Goa 403726, India +3International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru – 560089, India +4Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5 +5Department of Physics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1 +6Institute of Physics, University of Greifswald, 17489 Greifswald, Germany +(Dated: January 23, 2023) +Caustics are singularities that occur naturally in optical, hydrodynamic and quantum waves, +giving rise to high amplitude patterns that can be described using catastrophe theory. +In this +paper we study caustics in a statistical field theory setting in the form of the sine-Gordon model +that describes a variety of physical systems including coupled 1D superfluids. Specifically, we use +classical field simulations to study the dynamics of two ultracold 1D Bose gases (quasi-condensates) +that are suddenly coupled to each other and find that the resulting non-equilibrium dynamics are +dominated by caustics. Thermal noise is included by sampling the initial states from a Boltzmann +distribution for phononic excitations. We find that caustics pile up over time in both the number and +phase difference observables leading to a characteristic non-thermal ‘circus tent’ shaped probability +distribution at long times. +I. +INTRODUCTION +Wave focusing is ubiquitous in nature and leads to +localized regions of high amplitude called caustics that +dominate wavefields. +Everyday examples are provided +by rainbows and also the bright lines on the bottom of +water pools which are caused by the focusing of sunlight +by raindrops and surface water waves, respectively [1]. +Caustics also occur in water waves themselves as ship +wakes [2] and more dramatically as tsunamis (focused by +the topography of the seabed [3–5]) and tidal bores (fo- +cused by v-shaped bays [6]). Astrophysical examples in- +clude gravitational lensing by matter and the twinkling of +starlight due to time-dependent fluctuations in the den- +sity of Earth’s atmosphere. Natural focusing also leads +to the phenomenon of branched flow [7] and is speculated +to have given rise to the filamented nature of the large +scale structure of the universe [8–11]. In all these systems +caustics give rise to extreme amplitude fluctuations that +occur more frequently than those predicted by gaussian +statistics [12]. +A remarkable property of caustics is that they com- +monly take on particular characteristic shapes. This is +because caustics are singularities of the ray description, +i.e. they are places where two or more rays coalesce lead- +ing to a diverging intensity in the short wavelength limit +[13]. Such singularities are described by Thom’s catas- +trophe theory which rigorously shows that only certain +shapes of singularity are structurally stable against per- +turbations and hence occur under ‘natural’ or generic +∗ aagarw03@uoguelph.ca +† manas.kulkarni@icts.res.in +‡ dodell@mcmaster.ca +conditions [14–16]. +These special shapes or catastro- +phes form a hierarchy organized by dimension where the +higher ones contain the lower ones. Each member of the +hierarchy represents a class of equivalent shapes that can +be smoothly transformed into each other, but each class +is distinct and cannot be smoothly transformed into any +of the others. In two dimensions the only structurally +stable shape is the cusp and we shall see it appear fre- +quently when we plot quantities such as number fluctu- +ations versus time. +It is worth noting in this context +that the humble point focus that we associate with lens- +ing is structurally unstable and unfolds into an extended +caustic in the presence of perturbations (aberrations). +Natural lenses are of course never perfect and so typi- +cally produce the shapes predicted by catastrophe theory. +The upshot of all this is that caustics represent a form of +universality in nonequilibrium wave dynamics: they fall +into equivalence classes each with their own shapes and +scaling properties analogous to, but a generalization of, +equilibrium phase transitions [13, 17]. +Caustics should equally be present in quantum waves +where, due to the probabilistic interpretation, they cor- +respond to regions of high probability density. Quantum +matter wave caustics have been seen in experiments with +cold neutrons [18, 19], electron microscopes [20], atom op- +tics [21–23], and most recently in atom lasers [24]. The- +oretical works on such matter wave caustics have also +considered their ‘fine structure’ [13] which features a lat- +tice of vortices [25–27]. Quantum fields are another area +where caustics are expected to form naturally during dy- +namics. Early work centred on the electromagnetic field +[28, 29], including an interpretation of Hawking radiation +as a ‘quantum catastrophe’ [30], and more recently this +idea has been extended to quantum many-particle sys- +tems including bosonic Josephson junctions [26, 31, 32], +the XY model with long-range interactions (Hamiltonian +arXiv:2301.08410v1 [cond-mat.quant-gas] 20 Jan 2023 + +2 +Figure 1. Schematic of the setup we consider. The top fig- +ure shows two quasi one-dimensional gases that are prepared +independently and then suddenly coupled together. We call +this process of sudden coupling a “J-quench”. ρ1(z) and ρ2(z) +represent the density (red) in the first and second conden- +sates, respectively. Similarly, φ1(z) and φ2(z) represent the +phases (black) of the two condensates. Prior to the J-quench, +these fields in the two condensates are independent and con- +tain thermal fluctuations. The bottom figure shows how a J- +quench could be implemented by suddenly reducing the tun- +neling barrier height in a double well potential from a higher +to a lower value. +mean field model) [33], quantum spin chains [27] and the +Bose-Hubbard model [34]. +One point to appreciate is +that the caustics in many-body systems can occur in the +wavefunction associated with an entire N-body configu- +ration. Quantum many-particle caustics therefore live in +Fock space which can have a large number of dimensions +and hence lead to very complicated catastrophes [34]. +However, catastrophes obey projection identities which +means that when projected down to lower dimensions one +obtains either the same catastrophe or one lower down +the hierarchy [35]. Thus, low order correlation functions +obtained by integrating out most of the degrees of free- +dom will also generically contain caustics [27]. +In this paper we study caustics in the sine-Gordon (SG) +model. +The (classical) SG model obeys the nonlinear +wave equation +∂2φ +∂t2 − c2 +0 +∂2φ +∂z2 + ω2 +0 sin φ = 0 +(1) +where φ = φ(z, t) is a one dimensional field, and c0 and +ω0 represent a characteristic speed and frequency, respec- +tively. If c0 is taken to be the speed of light then Eq. (1) is +relativistically covariant, being a nonlinear version of the +Klein-Gordon equation and reducing to it when φ ≪ 1 +such that sin φ ≈ φ. The SG model received attention +from the high energy physics community in the 1970s due +its soliton solutions [36–39], but also describes the low en- +ergy physics of a considerable range of condensed matter +systems including crystal dislocations [40], domain walls +in magnetic [41] and binary superfluid [42] systems, the +Heisenberg spin chain with a field induced gap [43–45], +one-dimensional Bose gases in periodic potentials (that +can capture the Mott-insulator to superfluid transition +in one dimension) [46, 47], two-dimensional Bose gases +realizing the XY model [48], trapped ions [49], and two +tunnel-coupled one-dimensional Bose gases [50–57]. The +fact that the SG model is both nonlinear and integrable +means that attention is often focused on its soliton so- +lutions, but part of our mission in this paper is to point +out that these same properties also imply that caustics +(which are associated with the existence of tori in phase +space [58]) are expected to occur generically, and we are +aware of only one previous study of caustics in this model +[59]. +The particular physical realization we have in mind +for this paper is a system composed of two elongated +quasi-one dimensional Bose gases coupled by tunneling +along their length; the field φ(z, t) in Eq. (1) gives the +relative phase between the two quantum gases. +Quasi +one-dimensional Bose gases have been created in a num- +ber of experiments over the last two decades using tightly +trapped ultracold atoms, and the remarkable tunability +of these systems allows the strongly interacting Tonks- +Girardeau regime [60, 61], the weakly interacting quasi- +condensate regime [62–65], and also the crossover be- +tween the two [66, 67], to be reached. It is important to +note that, in accordance with the Mermin-Wagner theo- +rem [68], one-dimensional Bose gases do not undergo true +Bose-Einstein condensation at low temperature, unlike +three dimensional gases. Instead, they can form quasi- +condensates where density fluctuations are suppressed +but phase fluctuations remain [69, 70]. In this paper we +shall work in the weakly interacting regime and assume a +state of the system consisting of a quasi-condensate plus +small thermal fluctuations. +A system comprised of two coupled quasi-one dimen- +sional gases can be made by taking a single gas and +splitting it in two along its long axis by switching on an +elongated double well potential. This is the experimen- +tal protocol typically adopted in a series of experiments +conducted by the Vienna group [63, 71–77]. The com- +bination of almost complete isolation from the environ- +ment, long relaxation times and spatially resolved mea- +surements of phase and number difference make these +experiments ideal for investigating many-particle quan- +tum dynamics, including fundamental questions such as +whether and how closed quantum systems reach equi- +librium. The gas can be split slowly so that it always +remains close to equilibrium leading to number squeezed +states [78, 79] or it can be split rapidly, leading to a so- +called quantum quench which launches the system into a +nonequilibrium state. +In this paper we shall consider the opposite quench + +pi(2) +(2)d +P2(2) +p1(z)3 +where two one-dimensional gases are suddenly connected +together (see schematic representation in Figure 1). This +touches on rather fundamental considerations in quan- +tum mechanics since it describes the build-up of coher- +ence between two initially independent systems, and is +therefore related to the double-slit experiment for many- +particle systems [53, 80–83]. We shall refer to this as a +“J-quench” because J is often used to denote the cou- +pling strength between the two wells. In a simple two- +mode description of a bosonic Josephson junction, i.e. +one that assumes a single mode in each well without the +quasi-continuum of low energy longitudinal modes that +are present in highly elongated traps, such a quench is +predicted to result in a periodic collapse and revival of +the atom number distribution between the two wells [84– +86]. Essentially the same behavior, but π/2 out of phase, +occurs in the relative phase which is the conjugate vari- +able to number difference. In Refs. 26, 31, and 32 these +revivals are shown to be examples of quantum caustics +in a many-particle system. One of our main aims here +is to investigate what happens to these caustics in the +presence of the dispersive longitudinal modes present in +the SG model, and is part of a wider program attempt- +ing to understand the role of caustics in quantum many +particle dynamics [17, 26, 27, 31–34]. +Due to the difficulty of solving the fully quantum SG +model we take a semiclassical-style approach based on +classical field configurations which are solutions of Eq. +(1). +Each configuration is analogous to a single geo- +metric ray in optics and we include fluctuations by sum- +ming many configurations. The initial conditions for each +field configuration are randomly sampled from a Boltz- +mann distribution. +This approach is similar in spirit +to the truncated Wigner approximation (TWA) [87–92] +which includes quantum fluctuations around the classi- +cal field by summing many rays sampled from a quan- +tum probability distribution. The TWA has previously +been applied to one-dimensional Bose gases by Martin +and Ruostekoski [93, 94] who studied dark solitons, and +also to the connection problem of two zero temperature +one-dimensional Bose gases by Dalla Torre, Demler and +Polkovnikov [53], who proposed a universal scaling form +for the phase dynamics after the quench. More recently, +the TWA has been used by Horváth et al. [95] to study +the surprisingly sudden relaxation of the phase seen in +the Vienna BEC splitting experiments [77]. In this paper, +we include both the quantum fluctuations arising from +coupling two independent systems and thermal fluctua- +tions arising from thermal phonons in the longitudinal +modes and compare the time evolution of macroscopic +variables (the total number difference and phase differ- +ence) in the SG system against the simpler two mode +system [17, 26, 31]. +We find that following a quench +caustics dominate the dynamics of the macroscopic vari- +ables of both systems, even in the presence of thermal +fluctuations. Due to the singular nature of caustics, and +combined with their structural stability, we therefore pro- +pose that strong nongaussian fluctuations are a generic +phenomenon following a quench in the SG model (and +indeed, in integrable or moderately chaotic many-body +systems in general). +The caustics we discuss in this paper also have implica- +tions for the question of relaxation towards equilibrium at +long times in many particle systems. While chaotic (non- +integrable) and open quantum systems should thermalize +(although a complete description is still the subject of ac- +tive research [96–103]), closed integrable models do not +reach a conventional Gibbs state. We show here that in +the SG model there is a pile-up of caustics leading to a +singular shape for the long time probability distribution +for the macroscopic variables that resembles the shape of +a circus tent and is quite distinct from the thermal equi- +librium prediction. We find that an analytic approxima- +tion to the singular distribution based on an ergodic pen- +dulum (assuming a microcanonical or ‘equal-probability’ +distribution) provides a good fit to the numerical data. +The plan for the rest of this paper is as follows. We +start in Sec. II by deriving the SG hamiltonian from the +many-body description of two coupled 1D Bose gases. In +Sec. III we describe the natural length and time scales +and use them to write the SG hamiltonian and equa- +tions of motion in convenient dimensionless forms. Sub- +sequently, in Sec. IV we develop a method for finding the +initial conditions for the SG equations of motion. +We +assume that prior to the quench the two Bose gases are +independent and at thermal equilibrium with a bath at +temperature T. The initial conditions are obtained by +stochastically sampling the Fourier modes of a 1D quasi- +condensate obeying the Tomonaga-Luttinger liquid the- +ory. With the initial conditions in hand, in Sec. V we +give the main results of this paper which are the dy- +namics of the macroscopic number and phase difference +variables obtained by solving the equations of motion +numerically. In Sec. VI we consider the bigger picture +and examine the universal aspects of our results includ- +ing the influence of caustics on the coherence as well as +the long time dynamics and the establishment of (non- +thermal / non-Gaussian) equilibrium. +We conclude in +Sec. VII. There are also six appendices where we give +the details of the calculations as well as bench marking +our numerical method. +II. +FROM TWO COUPLED CONDENSATES TO +THE SINE-GORDON PLUS MODEL +We begin by deriving the SG model as an effective low +energy description for two coupled one-dimensional Bose +gases. For the sake of clarity, we list the main simplifica- +tions employed in this work: +• the treatment of a quantum many body problem +by a semiclassical method (TWA). +• the neglect of a weak harmonic trap along the +long axis which would otherwise lead to a non- +uniform longitudinal density (this can be avoided + +4 +in box traps which, although rarer, can be realized +[76, 104]) +• the assumption of a constant value for the tunnel +coupling J along the entire length of the gases +• the neglect of coupling to symmetric and higher +transverse modes. Some more involved theoretical +models do include these effects [56, 57]. +These simplifications are not expected to qualitatively +alter the main results of this work due to the structural +stability of caustics. In other words, caustics are known +to be robust to perturbations in both the Hamiltonian +and initial conditions. +A theoretical description of two ultracold quasi-one di- +mensional gases made up of bosonic atoms of mass m, +and held parallel to each other so that the atoms can +tunnel between them at rate J, can be obtained from the +following microscopic Hamiltonian [50, 51, 74] +ˆH = +� +j=1,2 +� L/2 +−L/2 +dz +� +− ℏ2 +2m +ˆψ† +j(z) ∂2 +∂z2 ˆψj(z) + U(z) ˆψ† +j(z) ˆψj(z) + g1D +2 +ˆψ† +j(z) ˆψ† +j(z) ˆψj(z) ˆψj(z) +� +− +� L/2 +−L/2 +dz ℏJ +� +ˆψ† +1(z) ˆψ2(z) + ˆψ† +2(z) ˆψ1(z) +� +. +(2) +The indices j = 1, 2 label the two gases and each is as- +sumed to be tightly trapped in the x and y directions +so that those degrees of freedom are frozen into their +ground states. Only the longitudinal degree of freedom +z in each gas is taken to be active. In experiments there +will usually be a weak longitudinal trapping potential +U(z), although as mentioned above for simplicity we set +it to zero and hence consider a uniform system of length +L with periodic boundary conditions. The quantum field +operator ˆψj(z) annihilates a particle at point z and to- +gether with its hermitian conjugate obeys bosonic com- +mutation relations [ ˆψj(z), ˆψ† +j′(z′)] = δjj′δ(z − z′). The +interaction constant g1D characterizes the effect of atom- +atom scattering within each gas on the longitudinal de- +gree of freedom and can be controlled both in magnitude +and sign either through Feshbach or confinement-induced +scattering resonances [105]. We note in passing that a +possible alternative physical realization of this problem +could be a spinor Bose gas in a single quasi-one dimen- +sional trap [106]. In fact, bosonic Josephson junctions +where the atoms are held in a single trap and two atomic +spin states are used for the two states have already been +realized experimentally [107]. +A weakly interacting three-dimensional Bose gas at ul- +tracold temperatures will undergo Bose-Einstein conden- +sation and can be described to high accuracy by a clas- +sical field approximation (Gross-Pitaevskii theory [108]). +In a quasi-one dimensional geometry quantum fluctua- +tions can still be small if the density is not too low, and +under these circumstances the gas can be treated as a +quasi-condensate where the quantum field operators are +replaced by classical fields [69, 109, 110] +ˆψj(z) → ψj(z) = +� +n1D + ρj(z) exp[iφj(z)] . +(3) +Here n1D = N/L is the background density where N is +the number of atoms in each gas (for simplicity we as- +sume an equal number of atoms N in each gas; the struc- +tural stability of caustics means that they are stable to +small differences in n1D between the two gases). ρj(z) +and φj(z) are the atom number density and phase fluc- +tuations at each point z, respectively. These are canon- +ically conjugate variables and can even be quantized in +a semiclassical regime such that they obey the commu- +tation relations [ˆρj(z), ��φj′ (z′)] ≈ δjj′δ(z − z′) in a coarse +grained sense [110]. However, in the present paper ρj(z) +and φj(z) will be purely classical fields subject only to +thermal fluctuations. +We can further decompose the fields into their sym- +metric and antisymmetric components +ρs(z) = ρ1(z) + ρ2(z) +2 +, +ρa(z) = ρ1(z) − ρ2(z) +2 +φs(z) = φ1(z) + φ2(z), +φa(z) = φ1(z) − φ2(z) . (4) +If the fluctuations are small ρa will be small whereas ρs +will be comparatively large. The particle-particle inter- +action energy will then typically cause the dynamics of +the symmetric modes to occur at higher energy than the +antisymmetric ones, and consequently we can ignore the +symmetric degrees of freedom as long as we restrict at- +tention to low energies [50, 55, 72, 75]. The Hamiltonian +purely describing the antisymmetric variables is (see Ap- +pendix A for details) +HSG+ = +� L/2 +−L/2 +dz +� +g1D ρ2 +a(z) + ℏ2n1D +4m +�∂φa +∂z +�2 ++ +ℏ2 +4mn1D +�∂ρa +∂z +�2 +− 2ℏJn1D cos φa(z) +� +. +(5) +We refer to this as the “sine-Gordon plus” (SG+) Hamil- +tonian because it includes an extra term (the third term) +in comparison to the standard SG Hamiltonian. +This +term involves gradients of density fluctuations and results + +5 +in an energy cost which automatically suppresses den- +sity fluctuations at small length scales. It is also worth +noting that including this term means that the density +and phase fluctuations [the second term in Eq. (5)] are +incorporated on an equal footing. +This is also in ac- +cordance with Gross-Pitaevskii theory which suppresses +density fluctuations with wavelengths below the healing +length [95] +ξh = +ℏ +√mg1Dn1D +. +(6) +However, when n1D is relatively large the third term is +naturally suppressed in comparison to the others and can +be dropped as long as the density gradients are small +leading to the SG Hamiltonian [55, 74] +HSG = +� L/2 +−L/2 +dz +� +g1D ρa(z)2 + ℏ2n1D +4m +�∂φa +∂z +�2 +− 2ℏJn1D cos φa(z) +� +. +(7) +The nonlinear piece in both Hamiltonians is the cosine +term which originates from tunneling between the two +wells and occurs in all Josephson junction type prob- +lems. +It provides an effective potential well for phase +configurations φ(z, t) that play the role of rays. In fact, +as we shall see in Section V, it acts as an (imperfect) lens +that focuses rays excited by the quench to form caustics. +For the sake of brevity, and when we deem no confusion +can arise, we will omit the ‘a’ subscript on antisymmet- +ric variables since we will not be dealing with symmetric +degrees of freedom. +The fact that the two fields φ(z) and ρ(z) form a conju- +gate pair means that their equations of motion are given +by Hamilton’s equations +˙φ = 1 +ℏ +δH +δρ(z) +˙ρ = −1 +ℏ +δH +δφ(z) +(8) +where H is the Hamiltonian density defined via +H = +� L/2 +−L/2 +H dz. +(9) +Applying these equations to the SG+ Hamiltonian given +in Eq. (5) we find the following of equations of motion +dφ(z, t) +dt += 2 g1D +ℏ ρ(z, t) + 2 +ℏ +4mn1D +∂2ρ(z, t) +∂z2 +dρ(z, t) +dt += 2 ℏn1D +4m +∂2φ(z, t) +∂z2 +− 2Jn1D sin[φ(z, t)] . +(10) +These are the key equations we use to solve for the dy- +namics of the field configurations. They have the form of +Josephson’s equations [111] augmented by second order +spatial derivatives ∂2φ/∂z2 and ∂2ρ/∂z2 which account +for phase and density fluctuations along the longitudi- +nal direction. Combined with the sine term, they will +cause wavepackets to disperse along z. In the absence of +these terms we have exactly the equations of motion for +a pendulum where φ is the angular displacement from +equilibrium and ρ plays the role of angular momentum. +The dependence on z suggests an interpretation in terms +of a continuous chain of many pendula each coupled to its +neighbors by the spatial derivative terms and is reminis- +cent of the Fermi-Pasta-Ulam-Tsingou problem [50, 112]. +In this paper the coupled equations of motion given in +Eq. (10) will be solved numerically for a system of length +L. To perform the numerical computations we discretize +the system on a spatial grid with NL + 1 points which +makes the grid spacing a = L/NL. The positions of the +grid points are given by z = ra where r is an integer +r = −NL +2 , . . . , NL +2 +(11) +and NL is chosen to be an even integer. +There is in fact a physical limitation on the grid size. +Eq. (10) is classical and valid only on length scales greater +that healing length ξh [51, 95]. Therefore, any numerics +performed on Eq. (10) are meaningful only when the lat- +tice grid size a is greater than ξh. +In particular, NL +should be such that a > ξh which implies +N 2 +L < mg1Dn1DL2 +ℏ2 +. +(12) +We fulfil the condition given in Eq. (12) in our numerics. +III. +NATURAL SCALES +Let us express the SG/SG+ Hamiltonians and equa- +tions of motion in terms of the natural scales for a one- +dimensional quantum fluid. For a length scale we chose +the healing length ξh given in Eq. (6). The ratio of the +healing length to the mean interparticle spacing 1/n1D +motivates the definition of the Luttinger parameter +K = +� +n1D(ℏπ)2 +4g1Dm . +(13) +This dimensionless quantity measures how strongly in- +teracting the system is - when K ≫ 1 the healing length +is much greater than the interparticle spacing and the +system is in the weakly interacting (quasi-condensate) +regime. Another key physical quantity is the speed of +sound +c = +�g1Dn1D +m +. +(14) +This can be used to define a characteristic energy, namely +that associated with phonons (quanta of sound) +E = ℏω = ℏc +ξh +(15) + +6 +where we have set the natural frequency ω to be the ratio +of the speed of sound to the healing length. +We therefore transform to the following dimensionless +variables +z −→ ˜z = z +ξh +, +t −→ ˜t = c +ξh +t +ρ −→ ˜ρ = ρ ξh +, +φ −→ ˜φ = φ +(16) +and defining ˜HSG = HSG/E and likewise for ˜HSG+ we +obtain the two Hamiltonians in dimensionless form +˜HSG = +� L/2 +−L/2 +d˜z +� +Γ ˜ρ2 + ϵ +� +∂ ˜φ +∂˜z +�2 +− 2J cos ˜φ +� +(17) +and +˜HSG+ = +� L/2 +−L/2 +d˜z +� +Γ ˜ρ2 + ϵ +� +∂ ˜φ +∂˜z +�2 ++ Γ +4 +�∂˜ρ +∂˜z +�2 +− 2J cos ˜φ +� +(18) +where the coefficients are given by +Γ = +π +2K , ϵ = K +2π , J = K +2π +ξ2 +h +ξ2s +. +(19) +In the last term we have introduced the spin healing +length +ξs = +� +ℏ +4mJ +(20) +which provides a measure for the distance over which +coherence between the two gases is restored due to the +tunnel coupling J [55]. At finite temperatures another +useful length scale is the thermal phase coherence length +λT = 2ℏ2n1D +mkBT . +(21) +The dimensionless form of the equations of motion can +now be given. For the SG model we find +d˜φ +d˜t = 2Γ˜ρ +d˜ρ +d˜t = 2ϵ∂2 ˜φ +∂˜z2 − 2J sin ˜φ +(22) +and for the SG+ model we obtain +d˜φ +d˜t = 2Γ˜ρ − Γ +2 +∂2˜ρ +∂˜z2 +d˜ρ +d˜t = 2ϵ∂2 ˜φ +∂˜z2 − 2J sin ˜φ . +(23) +IV. +INITIAL CONDITIONS +The dynamics we seek to study in this paper start from +a J-quench where two independent one-dimensional gases +at thermal equilibrium are suddenly coupled. In order +to obtain the initial density and phase fluctuations of +these gases we use the Tomonaga-Luttinger (TL) model +that provides the universal low energy effective theory for +one-dimensional systems (low energy limit of the Lieb- +Lininger theory, for example) [53]. +A. +Tomonaga-Luttinger (TL) liquid +In our notation the TL Hamiltonian reads +HTL = +� L/2 +−L/2 +dz +� +g1Dρj(z)2 + ℏ2n1D +4m +�∂φj +∂z +�2� +(24) +where j labels either of the two gases. We henceforth, +omit this label for the sake of brevity with the under- +standing that in this section the density and phase fields +refer to just one of the two gases. Eq. (24) has the same +mathematical structure as the SG model but without the +tunnelling term. +If we include density fluctuations we +find +HTL+ = +� L/2 +−L/2 +dz +� +g1Dρ(z)2 + ℏ2n1D +4m +�∂φ +∂z +�2 ++ +ℏ2 +4mn1D +�∂ρ +∂z +�2 � +. +(25) +The TL model is quadratic and hence its thermal fluc- +tuations can be treated exactly. To this end it is useful +to work in Fourier space and we apply discrete Fourier +transforms defined on the numerical grid with NL points +as discussed at the end of Section II. The phase field φ +and its Fourier transform ϕ are related by +φr = +1 +√NL + 1 +NL/2 +� +k=−NL/2 +ϕk exp +� +i 2πkr +NL + 1 +� +ϕk = +1 +√NL + 1 +NL/2 +� +r=−NL/2 +φr exp +� +−i 2πkr +NL + 1 +� +. +(26) +The discrete data {φr} = {φ−NL/2, . . . , φ0, . . . , φNL/2} +and its transform are located symmetrically about r = 0 +and k = 0, respectively. Since the value φr of the field +at each coordinate space grid point is a real number the +condition +ϕ−k = ϕ∗ +k must hold. Similarly the density +fluctuation field ρ and its Fourier transform ϱ are related +by +ρr = +1 +√NL + 1 +NL/2 +� +k=−NL/2 +ϱk exp +� +i 2πkr +NL + 1 +� +ϱk = +1 +√NL + 1 +NL/2 +� +r=−NL/2 +ρr exp +� +−i 2πkr +NL + 1 +� +(27) + +7 +where again the reality of the field in coordinate space +requires that ϱ−k = ϱ∗ +k. Inserting these transformations +in Eq. (25) we obtain (see Appendix B for details) +HTL+ = a g1D +NL/2 +� +k=−NL/2 +|ϱk|2 ++ a ℏ n1D +NL/2 +� +k=−NL/2 +ℏπ2k2 +mL2 |ϕk|2 ++ a +ℏ2 +4mn1D +NL/2 +� +k=−NL/2 +4π2k2 +L2 +|ϱk|2 . +(28) +Before proceeding with further analysis of Eq. (28), it is +worth noting that it can be recast in a standard Luttinger +liquid form +HTL+ = acℏ +2 +NL/2 +� +k=−NL/2 +�K +π +4π2k2 +L2 +|ϕk|2 + π +K |ϱk|2 ++ K +π +4π2k2 +N 2 |ϱk|2 +� +(29) +where the strength of the terms depends either on K or +1/K. +Applying the transformations given in Eq. (16), the +Fourier space variables can be written in dimensionless +form as +ϱk −→ ˜ϱk = ξhϱk +, +ϕk −→ ˜ϕk = ϕk +(30) +and the TL+ Hamiltonian given in Eq. (28) scaled by the +energy E = ℏc/ξh is given by +˜HTL+ = +˜L +NL +NL/2 +� +k=−NL/2 +�ϵ 4π2k2 +˜L2 +| ˜ϕk|2 + Γ|˜ϱk|2 ++ Γ π2k2 +˜L2 +|˜ϱk|2 +� +(31) +where ˜L = L/ξh is the ratio of the system size to the +healing length. Comparison with the spatial version of +HTL+ given in Eq. (25) shows where this factor comes +from: as the size is increased the range of the integration +increases linearly and this is accounted for by ˜L in the +Fourier transformed version. Note that all parameters +and variables in Eq. (31) are dimensionless. +B. +Thermal equilibrium +To find the initial conditions on the fields ρj(z) and +φj(z) we assume that each gas is at thermal equilibrium +such that the excitation (phonon) modes of the TL+ +Hamiltonian are populated with a probability given by +the Boltzmann distribution. The range of temperatures +we simulate is listed in Table I along with the values +of all the other key parameters, and is chosen so as to +correspond to realistic experimental conditions (the tem- +perature must be low enough that the quasi-condensate +description is valid). +In the canonical ensemble of statistical mechanics the +probability that a system at thermal equilibrium has +the phase space configuration s = q1, p1, q2, p2...qN, pN +is proportional to the Boltzmann weight exp[−βH(s)], +where β = 1/kBT and H = � +i p2 +i /2m + V (qi). +The +Hamiltonian in Eq. (31) is quadratic and hence the Boltz- +mann weight becomes that of a series of independent har- +monic oscillators +e− ˜β ˜ +HTL+ = +� +k +e−P 2 +k /2σ2 +ρ+ e−Q2 +k/2σ2 +φ+(k) +(32) +where ˜β = (ℏc/ξh)/kBT is the appropriately scaled tem- +perature parameter and we have introduced the real vari- +ables Qk and Pk which are related to the old variables +by +˜ϕk = Qkeiαk, +˜ϱk = Pkeiβk. +(33) +The phases αk and βk allow for the fact that ˜ϕk and ˜ϱk +can be complex numbers. The variances in Eq. (32) are +given by +σ2 +ρ+(k) = NL +2˜β +1 +Γ˜L(1 + π2k2/˜L2) +(34) +σ2 +φ+(k) = NL +2˜β +˜L +4π2k2ϵ . +(35) +The partition function can now be written down as +Z = +� +k +� ∞ +−∞ +e− ˜β ˜ +HTL+ dPkdQk += +� +k +� +σρ+ +√ +2π +� � +σφ+(k) +√ +2π +� +(36) +and hence the probability P of a particular configuration +(Q1, Q2, ...., P1, P2, ....) is +P = +� +k +� +e−P 2 +k /2σ2 +ρ+ +σρ+ +√ +2π +� � +e−Q2 +k/2σ2 +φ+(k) +σφ+(k) +√ +2π +� +. +(37) +This is seen to be the total probability distribution for +independent random variables Pk and Qk drawn from +normal distributions. Thus, the absolute values of the +Fourier coefficients ˜ϱk and ˜ϕk are normally distributed +random variables with zero mean and variances given by +Eqns. (34) and (35). We sample these numerically from +normal distributions to generate the initial system con- +figuration. The phases αk and βk given in Eq. (33) do +not appear in the Boltzmann weight and are chosen ran- +domly from the range [0, 2π). In fact, for both the phases +and the amplitudes we only need to choose the values for + +8 +terms with k ≥ 0 because the reality conditions imply +that we can put +Qk = Q−k , +Pk = P−k, +αk = −α−k , +βk = −β−k . +(38) +So far we have only considered the initial state of a sin- +gle gas. By subtracting the results for two gases we can +obtain the initial values of the antisymmetric variables +ρa(z) and φa(z) defined in Eq. (4). Actually, due to the +fact that the SG+ Hamiltonian with J = 0 and expressed +in terms of antisymmetric variables as given in Eq. (5) +formally has the same structure as the TL+ Hamiltonian +given in Eq. (25), sampling initial data for two gases is +unnecessary and one can obtain ρa(z) and φa(z) directly +by sampling them as though they were from one gas de- +scribed by the TL+ Hamiltonian. However, in doing so, +consideration needs to be given to the average value of +relative phase φa(z) because both the SG+ and TL+ +Hamiltonians only contain the spatial derivative of the +phase but not the phase itself. Its average value is there- +fore not determined by energy considerations and is left +to float freely. This is also apparent in the Fourier trans- +formed version of the TL Hamiltonian given in Eq. (31) +where the k = 0 term involving ˜ϕ0 is absent due to the +vanishing of its coefficient which is proportional to k2. +To take into account the random phase difference be- +tween the two gases one can chose ˜ϕ0 to be a random +number in the range [−π . . . π) but multiplied by a factor +of √NL + 1 in order to respect the normalization in Eq. +(26). This gives values of the average value of φa(z) in +the desired range −π and +π. +The random value of the initial phase difference is ac- +tually a key feature of the J-quench. It populates the +cosine potential landscape in the Hamiltonian with uni- +form probability. As the trajectories roll back and forth +in this potential they form caustics. +In effect, the co- +sine potential acts as an imperfect lens that focuses an +initially flat ‘wavefront’ over time. +C. +Choice of parameters +There are three constraints which must be satisfied in +order to have a quasi-one dimensional condensate [55]. +To ensure minimal scattering into the transverse modes +we need the interaction to be sufficiently weak which im- +plies µ = g1Dn1D ≪ ℏω⊥ where µ is the chemical poten- +tial and ω⊥ is the transverse trapping frequency. More- +over, the temperature needs to be low enough such that +transverse modes are not thermally excited leading to the +inequality kBT ≪ ℏω⊥. Finally, in order to have a quasi- +condensate which permits a semiclassical approach we +need weak interactions in comparison to the zero-point +kinetic energy associated with the density of the parti- +cles. This implies n1Dg1D ≪ ℏ2n2 +1D/m which means the +Symbol +Parameter +Value +ω⊥ +trapping frequency +2π × 3 kHz +m +mass of Rb atom +1.41 × 10−25 kg +as +scattering length +98 × 0.52 Å +N +number of atoms +1200 +L +system length +18 µm +n1D +average density +6.7 × 107m−1 +g1D +2 ℏascatω⊥ +2 × 10−38 Jm +K +Luttinger parameter +25 +T +temperature +2 - 20 nK +J +J-quench +0 - 30 Hz +NL +number of grid points +50 +c +speed of sound +3 × 10−3 m s−1 +a +grid spacing +0.36 µm. +ξh +healing length +0.24 µm +λT +phase coherence length +38 − 380 µm +ξs +spin healing length +2.5 µm +Table I. Table containing important parameters and their val- +ues. The parameters are chosen to be experimentally feasible +and correspond roughly to those reported in references [72– +77]. +Luttinger parameter should obey K ≫ 1. All the param- +eter values we use satisfy these three inequalities. +In quasi-one dimensional gases the interatomic inter- +action parameter g1D is related to the scattering length +as and transverse trapping frequency as g1D = 2ℏasω⊥. +For 87Rb atoms we have as ≈ 98 × 0.52 Å[113] and we +will assume ω⊥ = 2 π×3 kHz [77]. The full list of pa- +rameters used in our simulations is given in Table I and +roughly corresponds to those used in the experiments by +the Vienna group [72–77]. +For our numerical simulations we choose a grid size +that slightly exceeds the healing length because, as ex- +plained above, this cuts off unphysical density fluctua- +tions [51, 95]. This condition is given in Eq. (12) but can +be expressed succinctly in terms of Γ as N 2 +L < ΓN 2. The +magnitudes of ˜ρ and ˜φ also need to be considered. The +phase difference can take the full range +π to −π, but +the number difference is limited by the condition that +the total number difference (integrated over the entire +system) cannot exceed the total number of particles. In +fact, due to the random nature of sampled thermal fluc- +tuations, the integral of ˜ρ is always approximately zero. +However, the validity of the SG/SG+ model requires that +local density fluctuations be small in comparison to the +background density n1D, see Appendix A. Translated +into the scaled variables this means that at any point +˜ρ(˜z) ≪ n1Dξh. In practice we choose ˜ρ(˜z) ≤ 1.6 so that +the fluctuations are an order of magnitude smaller than +the background density. + +9 +D. +Examples of Initial conditions +In Figure 2 we present typical spatial profiles of the +initial number difference field ˜ρ (upper row) and phase +difference field ˜φ (lower row). Each profile provides the +initial conditions for a single classical field trajectory and +is obtained by summing up thermally activated phonons +(Fourier modes) using the Tomonaga-Luttinger model. +The different columns show the effect of changing tem- +perature T or Luttinger parameter K. +As expected, +when T is increased the fluctuations in both ˜ρ and ˜φ +increase. By contrast, if K is increased the maximum +magnitude and jaggedness of ˜ρ increases but the jagged- +ness of ˜φ decreases. Referring to Eq. (19) we can see that +this is because the coefficient multiplying the density fluc- +tuation term in the Hamiltonian is Γ = π/2K which de- +creases as K increases leading to increased variance of ϱk +modes according to Eq. (34). The phase fluctuation term +shows the opposite behavior because its coefficient in the +Hamiltonian (which only appears as the spatial gradient +of ˜φ) is ϵ = K/2π which increases as K increases and +this reduces the variance of the ϕk modes according to +Eq. (35), thereby making the ˜φ profiles smoother. +V. +NUMERICAL SIMULATIONS OF THE +DYNAMICS +In this section we explore the dynamics following a J- +quench. +Our approach is inspired by the TWA where +multiple classical field configurations are propagated in +time using the classical equations of motion, although in +our case the initial conditions are sampled from a ther- +mal distribution as described in Section IV rather than +a quantum distribution as in the standard TWA. +J-quench dynamics have previously been explored for +the simpler case of a two-mode zero temperature bosonic +Josephson junction where it was found that caustics dom- +inate the number and phase difference probability distri- +butions [17, 26, 31]. In the two-mode case it is possible +to compute the exact quantum dynamics for some thou- +sands of particles and compare them against the TWA. +The results (see Figure 1 in [31]) show good qualitative +agreement and give us confidence that the TWA can cap- +ture the main features of the quantum dynamics. Fur- +thermore, the inevitable presence of decoherence due to +the environment will tend to reduce the quantum dy- +namics to their classical limit (this has been investigated +in the two-mode case for a J-quench in [32]) increasing +the relevance of semiclassical calculations. In the present +work we are interested in whether the phonons along the +long axis disrupt or sustain these caustics. We will start +by reproducing the caustics presented in Ref. 31 for the +two-mode case and then add in the longitudinal modes +after that. +A. +Numerical Methods +The initial conditions are generated via random sam- +pling from Gaussian distributions. We then evolve the +equations of motion (Eq. 23 for the case of the full SG+ +model) using a Runge-Kutta solver with a user-defined +time step [114]. The endpoints of our system are treated +by imposing periodic boundary conditions. In Appendix +C we demonstrate the numerical convergence of the solver +by varying the temporal and spatial steps by tracking the +time evolution of the total energy (hamiltonian) which +should be a constant of the motion and obtain the fidu- +cial time and space resolution for all our calculations. +B. +Special case: two-mode approximation +In the two-mode approximation only a single mode in +each well is taken into account. This description is rele- +vant to the SG/SG+ model in the limit where the entire +length of each quasicondensate is perfectly synchronized +so that the fields ˜ρ(˜z) and ˜φ(˜z) do not depend on ˜z. +In this case the spatial derivative terms vanish and the +equations of motion in Eq. (23) reduce to +d˜φ +d˜t = 2Γ˜ρ +, +d˜ρ +d˜t = −2J sin ˜φ . +(39) +These are the standard Josephson equations of motion +and also correspond to those of a classical pendulum +[115]. Such synchronization can occur at very low tem- +peratures or when the coefficients ϵ and Γ are large +enough that they suppress spatial fluctuations in the ini- +tial conditions. +In Figure 3 we display the post-quench dynamics in +the two-mode approximation. +The left hand and cen- +tral panels show the time dependence of 150 indepen- +dent solutions of Eq. (39) which give the trajectories for +the number difference and phase difference, respectively. +Note that in this paper we use the color blue for tra- +jectories calculated within the two mode approximation +and reserve red for the trajectories of the full many mode +model. In accordance with our assumption that the two +wells start with an equal number of atoms, each solution +starts with ˜ρ = 0. +And as discussed in Section IV B, +the initial value of ˜φ is randomly chosen from the range +[−π, π) because the two condensates are independent be- +fore the J-quench. +The most striking feature of Figure 3 is the series of +cusp-shaped caustics that form in both variables. In or- +der to guide eye, we have have outlined the first cusp +caustic in the number difference variable using a black +curve (the calculation for this curve is given in Appendix +D). Like in optics, caustics are regions of high intensity +formed by the envelopes of families of rays (trajectories). +Each caustic is born at the centre of the distribution at +the tip of a cusp before spreading out in two arms that +move towards the edges of the distribution. +The fact + +10 +Figure 2. +Examples of initial spatial profiles of the number difference ˜ρ (top row) and phase difference ˜φ (bottom row). Each +profile is obtained by randomly sampling a thermal distribution using the method described in Section IV B, and each panel +includes ten different profiles. The parameter values common to all panels include the number of computational lattice points +NL = 50, grid spacing a = 0.36µm, and healing length ξh = 0.24 µm (the remaining parameters are listed in Table I). The +difference between the columns is as follows. The left column has the Luttinger parameter K = 25, and temperature T = 2 +nK giving a phase coherence length of λT = 380 µm. In the middle column K = 25, but the temperature is increased to 20 nK, +giving λT = 38 µm. In the right column, the value of K is artificially increased (without changing any other parameters) +to K = 250 and T = 2 nK. Increases in temperature excite stronger fluctuations in the profiles as expected. Increases in +the Luttinger parameter have opposite effects on ˜ρ and ˜φ. The maximum value and jaggedness of ˜ρ is increased whereas the +jaggedness of ˜φ is reduced. An explanation of this behavior is given in the main text. +that they are cusp shaped is in agreement with the pre- +diction of catastrophe theory that in two dimensions the +only structurally stable and hence generic singularities +are cusps. +Each trajectory represents a single experimental run. +The idea behind the TWA is that the number of tra- +jectories reaching a point ˜ρ at time ˜t is proportional to +the probability that a measurement of the true quantum +system would yield that value of ˜ρ. An equivalent inter- +pretation holds for the ˜φ trajectories. The caustics have +the highest probability density and hence give the values +most likely to be observed. Of course, if we only con- +sider the average values of ˜ρ or ˜φ we would get zero in +both cases due to the symmetry of the distributions and +hence miss the caustics. Many experimental runs must +be performed in order to obtain the probability distribu- +tion where these patterns live. +The mechanism underlying caustics can be understood +from a phase space perspective, as shown in the right +hand panel of Figure 3. Each dot gives the number and +phase difference at a particular time for a different ini- +tial condition. The red dots are the initial values which +lie in a horizontal line because at ˜t = 0 all trajectories +have ˜ρ = 0. As time evolves the dots rotate around the +origin: the green and blue dots show two successively +later times. However, the nonlinearity of the Josephson +equations means dots further from the origin rotate more +slowly and this leads to the formation of a spiral or whorl. +At places where the whorl has a vertical segment a range +of different solutions all have the same value of ˜φ and this +stationarity of the distribution with respect to changes in +the initial conditions is what generates a caustic, in this +case a ˜φ-caustic. +Conversely, horizontal segments give +rise to ˜ρ-caustics. +In the absence of nonlinearity the equations reduce to +those of a harmonic oscillator +d˜φ +d˜t = 2Γ˜ρ +, +d˜ρ +d˜t = −2J ˜φ +(40) +giving rise to rigid rotation in phase space and the forma- +tion of perfect focal points in the number and phase dif- +ference variables, as shown in Figure 4. However, these +perfect revivals of the initial state are not stable: any +nonlinearity will cause the focal points to evolve into the +extended cusp caustics shown in Figure 3. +The frequency of the linearized motion is known in +Josephson junction terminology as the plasma frequency. + +3 +2 +1 +Initial +0 +-1 +-2 +-3 +-20 +-10 +0 +10 +20 +Grid points (r)3 +2 +1 +Initial pr +0 +-1 +-2 +-3 +-20 +-10 +0 +10 +20 +Grid points (r)3 +2 +1 +Initial +0 +-1 +-2 +-3 +-20 +-10 +0 +10 +20 +Grid points (r)3 +2 +Initial $r +1 +0 +-1 +-2 +-3 +-20 +-10 +0 +10 +20 +Grid points (r)3 +2 +1 +Initial $r +0 +-1 +-2 +-3 +-20 +-10 +0 +10 +20 +Grid points (r)3 +2 +1 +0 +Initi: +-1 +-2 +-3 +-20 +-10 +10 +10 +20 +Grid points (r)11 +Figure 3. +Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and phase space distribution (right) following +a J-quench from J = 0 to J = 30 Hz in the two mode approximation governed by the Josephson equations given in Eq. (39). +The other parameter values are given in Table I. Each panel contains 150 trajectories: each trajectory starts with ˜ρ = 0 at time +˜t = 0 but has an initial phase randomly sampled from [−π, π). Both number and phase difference variables display a series of +cusp shaped caustics given by the envelopes of families of trajectories; to guide the eye we have outlined the first cusp caustic +in the ˜ρ variable with a black curve. In the right panel three different time slices of the results are plotted in phase space (˜ρ +versus ˜φ). Each dot corresponds to a different initial condition (trajectory) and the colors indicate the time: ˜t=0 (red), ˜t=50 +(green), ˜t=100 (blue). During time evolution the initial horizontal line winds into a whorl and the caustics in the ˜ρ and ˜φ plots +occur due to horizontal and vertical segments of a whorl, respectively. +Figure 4. +Dynamics of the number difference ˜ρ (left), phase difference ˜φ (middle), and the phase space distribution (right) in +the linearized version of the two-mode approximation [Eq. (40)] following a J-quench from J = 0 to J = 30 Hz. Like in Figure +3, there are 150 trajectories shown in each panel corresponding to different values of the initial value of ˜φ. However, in this +linearized case we obtain a series of perfect focus points (revivals of the initial state). This is because linearization gives rise to +rigid rotation in phase space without whorls. Unlike the extended cusp caustics seen in Figure 3 (which will be qualitatively +robust to details of the nonlinearity) perfect focus points are nongeneric because they are unstable to perturbations such as the +effects of nonlinearity. All parameter values and color labels are the same as Figure 3. +In our notation it reads +ωp = +√ +4ΓJ +(41) +and the period of the motion is therefore given by 2π/ωp. +For the case shown in Figure 4 we have Γ = 0.063 and +J = 0.037 giving a period ≈ 65. In fact, the tips of the +cusps in the nonlinear case also occur with this period +since they are formed from small amplitude trajectories +that only experience the quadratic bottom of the cosine +potential. +C. +General case: many-mode SG+ model +Simulations of the full SG+ model are shown in Figure +5, which represents one of the main results of this paper. +The trajectories in the left panel give the spatially av- +eraged number difference ⟨˜ρ(˜t)⟩z as a function of time +obtained by solving the equations of motion given in Eq. +(23) for the many-mode system and then averaging over +its length. The trajectories in the middle panel of Figure +5 give the equivalent spatial average of the phase differ- +ence ⟨˜φ(˜t)⟩z, and the right-hand panel is the phase space +picture. +Each trajectory is evolved from a single ran- +domly sampled field configuration (describing thermally +activated phonons) such as those shown in the top row +of Figure 2 and for the parameters given in Table I. We +observe that despite the inclusion of longitudinal modes +and the randomness of the initial conditions, the caustics +survive and are quite similar to those of the two-mode +approximation shown in Figure 3. +This suggests that +caustics are a generic feature of many particle dynamics + +1.5 +1.0 +0.5 +2Q +0.0 +-0.5 +-1.0 +-1.5 +.3 +-1 +0 +1 +2 +32 +1 +2Q +0 +-1 +-2 +0 +25 +50 +75 +5100 125 150 175 200 +2+3 +2 +1 +20 +0 +-1 +-2 +.3 +0 +25 +50 +75 +100 125 150 175 200 +2+2 +1 +2Q +0 +-1 +-2 +-3 +-1 +0 +1 +2 +w +iΦ2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +10 +25 +50 +75 +100 125 150 175 2003 +2 +1 +20 +0 +-1 +-2 +-3 +0 +25 +50 +75 +100125 150.175 20012 +Figure 5. +Dynamics of the spatially averaged number difference ⟨˜ρ⟩z (left), phase difference ⟨˜φ⟩z (middle), and phase space +distribution (right) for the full many-mode SG+ model following a J-quench from J = 0 to J = 30 Hz. Each panel contains +150 trajectories which are solutions of Eq. (23). The initial conditions are randomly sampled thermal phonons with the same +parameter values as those shown in the top row of Figure 2 and described in Table I. In particular, the number of numerical +lattice points is NL = 50 separated by a grid spacing of a = 0.36 µm, and the temperature is T = 2 nK. The healing length is +ξh = 0.24 µm, the spin healing length is ξs = 2.5 µm and the phase coherence length is λT = 380 µm. The different colors on +the phase space plot correspond to the same time slices as in the previous phase space plots. +following quenches, at least for systems whose underlying +physics is based on coupled nonlinear oscillators. Each +oscillator starts with a random phase and a noisy momen- +tum but the quench acts so as to give all the oscillators +a momentum kick at the same time ˜t = 0 leading to an +initial partial synchronization. As the system evolves in +time after the kick the different periods of nonlinear os- +cillators leads to cusp catastrophes in the distribution of +trajectories. If we had instead calculated only the expec- +tation values of the number and phase differences then +this underlying structure would not have been visible be- +cause it lives in the probability distribution rather than +the mean values. +A slice at fixed time through the probability distri- +bution for the spatially averaged phase variable ⟨˜φ⟩z is +shown in Figure 6. This is obtained by sorting the ⟨˜φ⟩z(˜t) +trajectories into bins each of which covers a small range +of ⟨˜φ⟩z and counting the number of trajectories in each +bin. The result is noisy due to the thermal fluctuations +but the caustics are clearly visible as strong peaks. These +peaks display the characteristic ‘square root’ divergence +of fold caustics [1] +P(⟨˜φ⟩z) ∝ +1 +� +˜φc − ⟨˜φ⟩z +(42) +where P(⟨˜φ⟩z) is the probability density and ˜φc is the +location of the caustic. The blue dashed lines in Figure 6 +are fits of Eq. (42) to the numerical data and we see that +the agreement is good. Although the height of the singu- +larities predicted by Eq. (42) is infinite at the caustic, this +function is integrable so that a probability distribution +with caustics is still normalizable (of course, the peaks in +the numerical data are of finite height because the num- +ber of trajectories is finite). A very similar pattern of +square root singularities at each caustic is obtained for a +time slice through the probability density for the number +difference variable so we shall not show it here. +Figure 6. +The probability density (red curve) as a func- +tion of ⟨˜φ⟩z obtained from the density of trajectories at time +˜t = 162 for the SG+ model. This corresponds to a slice at +fixed time through the middle panel of Figure 5, although +calculated using 10000 trajectories to improve the statistics +and averaged over a short time window of ∆˜t = 1 to remove +rapid time fluctuations. The red curve has been drawn with +a bin width d˜φ = 0.04 and is normalised such that the area +under the graph is 1. The caustics are clearly visible as di- +verging peaks and are well fitted (blue dashed curves) by the +inverse square root form given in Eq. (42) that is expected +for fold catastrophes [1] (the satellite caustics also have this +shape but the fit is not shown to avoid obscuring the data). +A very similar profile is obtained for the probability density +in the ⟨˜ρ⟩z variable (not shown). +D. +Effect of dispersion on the caustics +The double derivative terms in the SG+ equations of +motion given in Eq. 23 are responsible for transmitting +wave disturbances along the longitudinal axis and are not +present in the simpler two-mode case discussed in Section + +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +0 +25 +50 +75 +1001251501752003 +2 +1 +0 +-1 +-2 +-3 +0 +25 +50 +75 +100 125 150 175 200 +2+1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +0 +1 +2 +3 +()z1.0 +0.8 +M 0.4 +0.2 +0.0 +-3 +-1 +0 +1 +2 +3 +(0)z13 +V B. Initial thermal fluctuations in the SG+ model will +therefore disperse in z over time and it is interesting to +see what difference this makes to the caustics; comparison +of Figures 3 and 5 suggests it makes little difference to +spatially averaged variables. However, this observation is +for only one choice of the parameters ϵ and Γ that govern +the size of the derivative terms and also for relatively +short times. In particular, in Figure 5 the parameters +are ϵ ≈ 4 and Γ ≈ 0.06 which were chosen to match +experimental values [72–77]. In Figure 7 we compare the +long time dynamics of the two-mode approximation and +the SG+ model for the case where ϵ in the SG+ model +has been artificially increased by a factor of 10 (without +changing any other parameters), thereby increasing the +effect of spatial dispersion. Apart from this change, the +initial conditions and J-quench are similar to those used +in Figure 5. Note that we only use this increased value +of ϵ for the time propagation and not for the generation +of the thermal initial conditions. This avoids changing +the starting phase fluctuations from those used in Figure +5 which would otherwise be energetically suppressed and +would also lead to significantly different dynamics but is +not the comparison we would like to make here. From +Figure 7 we see that the strong coupling of neighboring +‘pendula’ does seem to largely wash out the caustics at +long times in comparison to the dispersionless two-mode +case, although some faint structure is still present which +underlines the structural stability of caustics. The long +time behavior will be further analyzed in Section VI. +E. +Effect of J on the caustics +Another parameter that affects the dynamics is the +tunnel coupling strength J [or its dimensionless version +J which is defined in Eq. (19)] that becomes non-zero +after the quench. The quench itself creates a strongly +nonequilibrium phase difference where all values of ˜φ are +equally probable independently of the value of J by virtue +of the fact that before the quench there is no phase co- +herence between the two quasicondensates. However, J +does control the post-quench dynamics. One way it does +this is via the frequency of the Josephson oscillations. +The cusps occur with a frequency given by the plasma +frequency in Eq. (41) which goes as +√ +J. +In Figure 8 we examine the effect of quenching to dif- +ferent J values, with the value of J increasing from left +to right. We can see the expected increase in frequency. +The amplitude of the motion also increases with J be- +cause immediately after the quench each trajectory finds +itself at a random point on the cosine potential energy +surface whose depth between valley top and valley bot- +tom is 2J . The initial potential energy of a field config- +uration is therefore −2J ⟨cos ˜φ0⟩z, where ˜φ0 is the phase +field ˜φ(˜z, ˜t) at the initial time. This configuration evolves +under the full Hamiltonian and upon spatial averaging is +seen to execute oscillations about the potential minimum. +The upper row in Figure 8 plots the spatially averaged +Figure 7. +Comparison of the long-time behavior of the phase +difference in the two-mode approximation (upper) and many- +mode SG+ model (lower). Both panels contain 150 different +runs and the initial conditions and J-quench are similar to +those of Figure 5 except that ϵ has been artificially multiplied +by 10 (without changing any other parameters) in the lower +panel. This enhances the effect of the spatial derivative term +in φ in the SG+ model (this term does not appear in the two +mode model). We see that in the upper panel the caustics +are still visible. By contrast, the stronger spatial interaction +causes dispersion and makes the caustics much less visible in +the lower panel. +number difference and according to Eq. (18) the maxi- +mum amplitude this can have is +⟨˜ρ⟩max +z += +� +2J (1 − ⟨cos ˜φ0⟩z) +Γ +(43) +where we have ignored the effects of spatial coupling (sec- +ond order derivative terms). Thus, ⟨˜ρ⟩max +z +also scales as +√ +J, and this is in correspondence with Figure 8. +The lower row of Figure 8 shows the behavior in phase +space. In these figures we have also included the unaver- +aged data, i.e. the ˜ρ and ˜φ values of each grid point at +the three selected times. This gives a sense of the size +of the statistical fluctuations due to the spatial degrees + +3 +2 +1 +0 +-1 +-2 +-3 +600625650675700725 750775800 +2+3 +2 +1 +0 +-1 +-2 +-3 +600 625 650 675 700 725 750 775 800 +2+14 +Figure 8. +Effect of quench strength J for J = 0 Hz, 3 Hz, and 30 Hz (from left to right). The top row shows the dynamics +of ⟨˜ρ⟩z with initial conditions sampled in the same way as in Figure 5. The bottom row plots the corresponding phase space +distributions. Like in previous figures, the different colors give different time instants: ˜t=0 (red), ˜t=50 (green), ˜t=100 (blue). +The dots with intense colors are the spatially averaged values. We have also included the raw data (without spatial averaging) +as faint dots. This gives an idea of the size of the statistical fluctuations due to the thermal initial conditions and is the same +for all values of J. In the left column there is no coupling between the two quasicondensates and hence no time evolution of +the spatially averaged data (the intense red, green, and blue dots sit on top of each other) although there can be evolution +of unaveraged data due to intrawell dynamics, i.e. without the J term in Eq. (10). As we increase the magnitude of J time +evolution leads to whorls with a greater vertical extent because more energy can be extracted from the cosine potential in Eq. +(18) giving larger values of ⟨˜ρ⟩max +z +. +of freedom. In the left hand column J remains zero for +all time and the only dynamics that can occur is along +the long-axis of each quasicondensate individually. The +middle and right hand panels, which have J = 3 and +J = 30 Hz, respectively, have the same initial statistical +fluctuations as the left hand one because, as mentioned +above, the initial distribution is set by the pre-quench +thermal fluctuations in the two quasicondensates and is +independent of J. However, as time evolves the effects of +J described by Eq. (43) become apparent because larger +J allows a greater value of ⟨˜ρ⟩max +z +and this stretches the +distribution along the vertical direction in comparison to +a smaller value of J. For a whorl to become apparent +⟨˜ρ⟩max +z +should at least exceed the width of the statistical +fluctuations and becomes better and better defined as J +is increased. +VI. +UNIVERSALITY AND CAUSTICS +We have already discussed the relationship between +nonlinearity and caustics in the preceding section. +As +motivated earlier, and expounded in Refs. 17, 26, 31, 33, +and 34, caustics also have implications for the universal +dynamics of quantum systems. We explore a few of these +effects in this section. +A. +Long time distribution: the circus tent +The quench generates collective excitations that lead +to caustics as shown in Figures 3 and 5 for the two non- +linear models (two mode and SG+) discussed above. The +caustics are born at the center of the probability distri- +bution (in either the ˜ρ or the ˜φ variable) at intervals of +the plasma period and move out to the edges over time. +Figure 6 plots the probability distribution for the SG+ +model as a function of ⟨˜φ⟩z at an intermediate time where +four pairs of fold caustics are discernible and shows how +they diminish in strength but are still present as they +move to the edges. The question then naturally arises +as to what happens at long times ˜t → ∞ when the dis- +tribution comprises of a large number of caustics and +whether it tends to a characteristic shape? The answer +is yes, and is shown in Figure 9 which is made in the +same way as Figure 6 but this time by calculating the +density of ⟨˜ρ⟩z trajectories and averaging over a time +window extending between ˜t = 800 and ˜t = 980 in order + +2.0 +1.5 +1.0 +0.5 +z(g) +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +0 +25 +50 +75 100 125 150 175 200 +t2.0 +1.5 +1.0 +0.5 +z(d) +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +0 +25 +50 +75 100 125 150 175 200 +t2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +0 +25 +50 +75 +100 125 150 175 2002.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +-3 +-2 +-1 +0 +1 +2 +3 +(0)z2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +-3 +-2 +-1 +0 +1 +2 +3 +(0)z2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +-3 +-2 +-1 +0 +1 +2 +3 +(0)z15 +to remove rapid fluctuations. The probability distribu- +tion takes a shape reminiscent of a ‘circus tent’ or ‘big +top’ and can be understood as follows. +The strongest +singularities present are the cusp tips born at the center +of the distribution which leads to this being the highest +point. Each cusp then splits into two fold arms (which +according to catastrophe theory are lower singularities) +that move outwards, reducing in height as they go, before +accumulating at the edges where there is a sharp drop to +zero. The position of the outer edge is set by the maxi- +mum energy that can be extracted from the quench and +is given by Eq. 43. +An analytic expression for the circus tent distribution +is given by the integral +PCT(˜ρ) = +1 +2πB +� 1 +˜ρ2/B2 +U(m, ˜ρ) +K(m) +dm +(44) +where +U(m, ˜ρ) = +1 +� +m(1 − m)(m �� ˜ρ2/B2)(1 + ˜ρ2/B2 − m) +, +(45) +K(m) is the complete elliptic integral of the first kind, +and B = 2 +� +J /Γ. This expression is plotted in Figure +9 as the dashed line and is derived in Appendix E un- +der the assumption that at long times we can model the +system by an ensemble of independent pendulua where +each pendulum is ergodic. In other words, each pendu- +lum obeys a microcanonical distribution where there is +equal probability for it to be found anywhere on its en- +ergy shell. The nature of the J-quench is such that it +leads to an ensemble with an equal probability for any +starting angle (this is different to an equal probability +for each energy due to the dependence of the density of +states on angle). As can be seen from Figure 9, PCT(˜ρ) +gives a good fit to the numerical data generated by both +the SG+ and two-mode models considered in this paper. +In Figure 9 we also include the thermal probability +distribution +PT (˜ρ) = 1 +Z +� ∞ +0 +PE(˜ρ) e−E/T D(E) dE +(46) +describing an ensemble of pendula at thermal equilibrium +at temperature T where PE(˜ρ) is the probability distri- +bution at fixed energy E, D(E) is the density of states +and Z is a normalizing factor. The details of our cal- +culation of PT (˜ρ) are given in Appendix F, where, for +example, PE(˜ρ) is given in Eq. (F2). The temperature +of this distribution is chosen such that the mean energy +of the thermal distribution ⟨E⟩T is equal to the mean +energy of the states excited by the quench. For a quench +to J = 30 Hz we show in Appendix F that the effective +temperature is 5.4 nK. +Clearly, the thermal distribution is very different to the +circus tent distribution: the thermal distribution takes +the form of a smooth gaussian with wings that extend +beyond ⟨˜ρ⟩max +z +because the thermal Boltzmann factor +Figure 9. +The long time probability distribution for the +number difference ˜ρ. The data points are from the different +nonlinear models considered in this paper averaged over the +spatial coordinate z and also over a time window ranging from +˜t = 800 to ˜t = 980 to remove fluctuations. The pink dashed +line is the circus tent distribution PCT given in Eq. (44) and +derived in Appendix E under the assumption of ergodicity; +the circus tent shape is due to the proliferation of caustics +at long times and gives a good fit to the data. +The solid +black curve is the thermal distribution PT with a temperature +chosen so that the expectation value of the energy matches +that provided by the quench. +allows for excitations with any energy (albeit with ex- +ponentially small probability) including those involving +pendula undergoing rotation as well as libration, whereas +the J-quench only excites librational motion. The proba- +bility distribution for a thermal pendulum is in fact quite +delicate to compute because of the singularity in the den- +sity of states between libration and rotation but the com- +bined result is smooth; see Appendix F for more details. +B. +Structural stability of caustics +The defining characteristic of the singularities de- +scribed by catastrophe theory is structural stability +against perturbations and this ensures that they occur +generically. The same is not true of isolated singularities +as can be seen by comparing Figures 3 and 4 where it +is shown that point foci do not survive the introduction +of nonlinearity. In two dimensions cusps are the unique +structurally stable catastrophe and from Figures 3 and 5 +we see that cusp-shaped caustics are indeed stable against +random thermal fluctuations. However, thus far we have +imposed the symmetrical starting condition that the ini- +tial number difference between the two quasicondensates +is zero. One may therefore wonder whether the caustics +we see are a consequence of this symmetry. To check that +this is not the case we show in Figure 10 the dynamics +for the case where the initial background density n1D in +the two quasicondensates differs by 10%. We see that + +1.0 +PcT +Thermal +two-mode +0.8 +many-mode SG+ +t +0.6 +0.4 +0.2 +0.0 +-2.0-1.5-1.0-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +(p)z,t16 +Figure 10. +Structural stability of caustics: here we investigate the effect of unbalanced densities on caustics by tracking the +same SG+ model dynamics as those shown in Figure 5 except for an initial density imbalance of 0.1 in the background of ˜ρ +at each point z. We see that the cusp caustics in the plots of ⟨˜ρ⟩z and ⟨˜φ⟩z versus time are distorted but still maintain their +basic structure. This is because the whorl in phase space is left intact despite having a displaced centre. Caustics are resilient +against imperfections and perturbations and we expect them to be present under realistic experimental conditions. +although the caustics in both ⟨˜ρ⟩z and ⟨˜φ⟩z are distorted +they maintain their basic cusp shape. Furthermore, the +phase space whorls still occur and this guarantees the +existence of caustics. +C. +Coherence factor and relaxation towards +equilibrium +Cold atom experiments have the ability to measure cor- +relation functions in nonequilibrium many-body states +[74, 116–118]. As a simple example let us consider the +coherence factor +C(˜t) = +� +⟨cos ˜φ⟩z +� +(47) +which depends on the spatial average of the phase dif- +ference field ˜φ(˜z, ˜t) between points along the two qua- +sicondensates. The outer brackets indicate an ensemble +average which means averaging over many trajectories +each sampled from the thermal distribution discussed in +Sec. IV. In the Vienna experiments, where one quasicon- +densate is suddenly split into two, the coherence starts +near unity and decays over time as the two quasiconden- +sates decohere [76, 77]. In the opposite case, where two +independent quasicondensates are suddenly coupled, one +expects the converse where the coherence starts at zero +and grows. This situation has been previously modelled +by Horváth et al. using both the TWA and a truncated +conformal space approach [95]. +They found that C(˜t) +initially grows and then undergoes damped oscillations +as it settles down towards a finite constant value. The +coherence factor therefore provides a measure of how the +system reaches equilibrium. In this context we note that +C(˜t) actually corresponds to an ensemble average of the +cosine term in the SG/SG+ Hamiltonian and thus gives +information on the exchange of energy between the dif- +ferent parts. In other words, since the total energy is a +constant of the motion, if the ‘potential’ part of the en- +ergy settles down to a constant this suggests the ‘kinetic’ +parts of the energy are also constant, at least from an +ensemble averaged point of view. Our aim in this sec- +tion is to see if the dynamics of C(˜t) is connected to the +caustics. +In Figure 11 we plot C(˜t) for two models: the full +SG+ model which is many-mode and nonlinear and a +linearized version which obeys the equations of motion +d˜φ +d˜t = 2Γ˜ρ − Γ +2 +∂2˜ρ +∂˜z2 +d˜ρ +d˜t = 2ϵ∂2 ˜φ +∂˜z2 − 2J ˜φ. +(48) +This differs from the linearized two-mode approximation +defined by Eq. (40) because it describes an elongated +multi-mode system. From Figure 11 we see that C(˜t) for +the SG+ model (dark blue curve) does indeed initially +grow, undergo damped oscillations and settle down to a +non-zero value (the fact that C(˜t) ̸= 0 at ˜t = 0 is due +to random fluctuations in the initial conditions: as we +include more trajectories we find that the initial value +gets smaller). Meanwhile, C(˜t) for the linear model (red +dashed curve) executes undamped oscillations and hence +does not settle down to equilibrium. Both models agree +during the first oscillation but strongly differ after that. +It is clear that nonlinearity is important for reaching +equilibrium at least as far as global quantities such as +C(˜t) are concerned. +We can understand this by inter- +preting the SG+ model as describing a chain of coupled +pendula. The nonlinearity of each pendulum means that +its period depends on the amplitude of its motion and +hence an ensemble of pendula whose motion is initiated +together by the quench, but all with different degrees of +excitation, will dephase from one another over time so +that collective oscillations are damped out. By contrast, +linear oscillators have a period independent of their am- +plitudes of motion and hence remain in phase. +Apart from the ensemble averages shown by the darker +curves in Figure 11, we have also included the individ- +ual trajectories for ⟨cos ˜φ⟩z as fainter curves. The linear + +2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +0 +25 +50 +75 +100 125 150 175 2003 +2 +1 +0 +-1 +-2 +-3 +0 +25 +50 +75 +100 125 150 175 200 +2t2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +-3 +-2 +-1 +0 +1 +2 +3 +(0)z17 +Figure 11. +The two dark lines give the time evolution of the +coherence factor C(˜t) defined in Eq. (47) for a linear model +(dashed-dotted red) and the SG+ model (solid blue). Both +models are multi-mode (many longitudinal modes along ˜z) +but the SG+ model is nonlinear. Also included as faint lines +are the raw trajectories ⟨cos ˜φ⟩z from which C(˜t) is composed. +As everywhere in this paper, ⟨. . .⟩z indicates a spatial aver- +age. +This figure highlights that recurrences present in the +linear case are suppressed by nonlinearity in the SG+ sys- +tem. +The ensemble average over trajectories with different +periods causes C(˜t) to relax towards an equilibrium value in +the case of the SG+ model in line with previous experimental +observations [76, 77] and theory [95]. +model displays harmonic motion and hence perfect re- +vivals whereas the trajectories in the nonlinear model +give rise to half-cusp caustics. +These caustics overlap +in time such that averaging over them causes the coher- +ence to strongly relax after a single period. It is not so +much that the caustics cause the relaxation, but rather +that both have a common origin in the nonlinearity of +the model and hence are generic features of dynamics in +complex systems. +VII. +SUMMARY AND CONCLUSIONS +The sine-Gordon (SG) model is a nonlinear integrable +field theory that can be used to describe a wide range +of systems from high energy physics to condensed matter +physics. A series of landmark experiments using two cou- +pled 1D atomic quasicondensates [63, 71–77] have real- +ized the SG model in a controllable quantum many body +environment. The key parameters can be varied in time +allowing the implementation of sudden quenches that ex- +cite many modes leading to nonequilibrium dynamics. +This is the setting we adopt for the current paper where +we use experimentally realistic parameters and compute +the dynamics of the number and phase difference fields. +However, in contrast to the usual experimental protocol +where the tunnel coupling J is suddenly switched off, +we consider quenches where it is suddenly switched on. +While the former case is adapted to studying dephasing, +decay and thermalization between the two subsystems, +the many body dynamics is governed by the Tomonaga- +Luttinger Hamiltonian describing independent 1D quasi- +condensates. If instead J is suddenly switched on then +the dynamics is that of the full SG model. +Our calculations employ a thermal version of the +semiclassical truncated Wigner approximation (TWA) +method. More specifically, we propagate a large num- +ber of classical field configurations over time with initial +conditions sampled from a distribution at thermal equi- +librium. The time evolved configurations (trajectories) +can be summed to obtain the probability distributions +for the observables and we find that these are dominated +by singular caustic patterns. The natural mathematical +description of caustics is catastrophe theory that predicts +a hierarchy of structurally stable singularities with char- +acteristic shapes that depend on dimension. In two di- +mensions (e.g. number or phase difference versus time) +the structurally stable catastrophes are fold lines that +meet at cusps. This is exactly what we find in both the +number and phase differences following a J-quench, see +Figure 5. +The probability distributions develop trains +of caustics that are born periodically as cusp points (lo- +cated at the center of the distribution if there is no tilt) at +each plasma period and evolve into pairs of fold lines that +gradually move out to the wings where they accumulate. +Fold catastrophes manifest as strong non-gaussian fluc- +tuations in the form of inverse square root divergences in +the intensity (probability density), as shown in Figure 6. +A special case is provided by the dynamics of a two +mode system as shown in Figure 3. +Here the equa- +tions of motion are the Josephson equations given in +Eq. (39). The only fluctuations we include in this ex- +ample are the quantum fluctuations in the initial rela- +tive phase between the two condensates as mandated by +the uncertainty principle applied to systems in relative +number eigenstates. +The two-mode case is relevant to +small systems where the higher modes are well above +the temperature scale and so any spatial fluctuations are +suppressed. By contrast, the many-mode case shown in +all the other figures includes both quantum fluctuations +and thermal fluctuations in the longitudinal modes, i.e. +thermal occupation of phonon modes in the 1D quasicon- +densates. Despite the presence of the many longitudinal +modes (typically 50 in our calculations, as set by the pa- +rameter NL) which give rise to highly random looking +phase and density profiles as seen in Figure 2, we find +that number and phase caustics survive for experimen- +tally realistic parameters. Furthermore, the qualitative +features of the caustics are stable against variations in +quench strength and density imbalance, as seen in Fig- +ures 8 and 10, respectively, and also against the details +of the model (in this paper we use the SG+ model which +augments the SG model by including longitudinal den- +sity gradients). All of these different examples confirm +the structural stability of caustics which is the reason why +they occur universally without the need for fine tuning. + +1.00 +0.75 +0.50 +(cos((z))z +0.25 +0.00 +-0.25 +-0.50 +-0.75 +Linearmany-mode +-1.00 +Non-linear many-mode SG+ +0 +50 100 150 200 250 300 350 400 +t18 +The proliferation of caustics over time combined with +their migration to the edge of the probability distribution +has important consequences for the long time probability +distribution. It takes on the shape of a circus tent featur- +ing a strong central peak due to the cusp tips which are +the most singular part of a caustic, flatter intermediate +regions, and rapidly decaying edges where the caustics +pile up, see Figure 9. This shape is quite distinct from a +gaussian thermal distribution and can be derived assum- +ing an ergodic hypothesis in which individual pendula +have equal probability to be anywhere on their energy +shell (see Appendix E). The approach to this equilibrium +distribution can be tracked over time using the coherence +factor (Figure 11) which is a spatial and ensemble average +over the phase field and corresponds to the cosine term in +the Hamiltonian if the latter is ensemble averaged. The +attainment of equilibrium relies on the nonlinearity of the +system to dephase itself when ensemble averaged. The +caustics also rely on the nonlinearity without which they +would reduce to nongeneric perfect revivals (point foci). +In this sense caustics are mutually exclusive to recur- +rences, at least in the statistical sense in which caustics +appear in this paper. +Caustics in the SG model could be observed experi- +mentally by measuring the probability density for either +the phase difference or the number difference. For ex- +ample, the phase difference can be obtained by releasing +the two quasicondensates from their double well potential +and letting them overlap [80–82]. This process must be +repeated many times and for as near identical initial con- +ditions and time evolution as possible in order to build +up a probability distribution, although due to the struc- +tural stability of caustics they will not be particularly +sensitive to differences in the experimental setup from +run to run. If the probability distribution is obtained for +a single time then we expect to see something like that +shown in Figure 6. In order to observe the time evolu- +tion of a caustic, one must then repeat the whole process +for a range of different evolution times. This is laborious +but technically possible, and since the first cusp caustic +appears at half the plasma period the experiment does +not need to run for long. +The singular nature of caustics means that they dom- +inate wave fields and are well known in hydrodynam- +ics and optics through phenomena such as tsunamis and +gravitational lensing. The results of this paper show that +they also occur in the nonequilibrium dynamics of 1D su- +perfluids where a quench plays an analogous role to an +underwater earthquake by generating strong excitations +beyond the linear regime that are focused in this case by +the cosine term in the SG Hamiltonian. The universal +properties of catastrophes imply caustics likely also occur +in the post-quench dynamics of other condensed matter +systems too: systems with more degrees of freedom will +display higher catastrophes beyond folds and cusps such +as hyperbolic and elliptic umbilics [34]. However, a spe- +cial feature of the SG model is that it is integrable and +so one may ask if that property plays a crucial role in +the existence of caustics. In this context, we note that +in classical mechanics caustics are closely associated with +the existence of tori in phase space upon which trajec- +tories live [58]. Tori are broken up by chaos, and thus +caustics are not expected to survive for long in systems +which are deep in the chaotic regime. Despite this, the +Kolmogorov-Arnold-Moser (KAM) theorem shows that +some tori survive in moderately chaotic systems [119], +which suggests caustics may also survive in cases where +the classical phase-space is mixed, which is the typical +case. Indeed, they survive in the three site Bose-Hubbard +model [34] which is known to be chaotic [120]. The im- +portant problem of extending the KAM theorem to quan- +tum mechanics [121] is thus intertwined with the analysis +of caustics in quantum systems and provides an interest- +ing direction for extending the present work. +ACKNOWLEDGEMENTS +We thank Ryan Plestid for contributions on ther- +mal field sampling in the early stages of this project, +Josh Hainge for suggesting the term ‘circus tent’, and +Igor Mazets for correspondence and advice about ex- +periments. +This work was supported by the Mitacs +Globalink research internship, by the Natural Sciences +and Engineering Research Council of Canada (NSERC), +and Research at the Perimeter Institute is supported +in part by the Government of Canada, through the +Department of Innovation, Science and Economic De- +velopment Canada, and by the Province of Ontario, +through the Ministry of Colleges and Universities. M.K. +would like to acknowledge support from the project +6004-1 of the Indo-French Centre for the Promotion of +Advanced Research (IFCPAR), Ramanujan Fellowship +(SB/S2/RJN-114/2016), SERB Early Career Research +Award (ECR/2018/002085) and SERB Matrics Grant +(MTR/2019/001101) from the Science and Engineering +Research Board (SERB), Department of Science and +Technology (DST), Government of India. M.K. acknowl- +edges support from the Infosys Foundation International +Exchange Program at ICTS. M.K acknowledges support +of the Department of Atomic Energy, Government of In- +dia, under Project No. 19P1112R&D. +Appendix A: Derivation of the sine-Gordon +Hamiltonian +In this appendix we derive the Hamiltonian HSG as +the effective low energy description of two cigar shaped +tunnel-coupled quasicondensates [50, 74] within a clas- +sical field description (Gross-Pitaevskii theory). Along +the way we also obtain a slightly enhanced Hamiltonian +HSG+ that includes contributions from the gradient of +density fluctuations that are not included in the sine- +Gordon (SG) Hamiltonian. These contributions are not +very important for our parameters but play an impor- + +19 +tant conceptual role by introducing an energetic price +for a rapidly varying density and hence effectively cut off +these fluctuations. +Assuming tight radial trapping such that each quasi- +condensate is in its radial ground state, meaning that +only longitudinal excitations are taken into account, the +second quantized Hamiltonian for the total system be +written +H = +� ∞ +−∞ +dz +� � +j=1,2 +� +− ℏ2 +2m +ˆψ† +j(z)∂2 ˆψj(z) +∂z2 ++ +U(z) ˆψ† +j(z) ˆψj(z) + g1D +2 +ˆψ† +j(z) ˆψ† +j(z) ˆψj(z) ˆψj(z) +� +− ℏJ +� +ˆψ† +1(z) ˆψ2(z) + ˆψ† +2(z) ˆψ1(z) +�� +. +(A1) +The quantum field operator ˆψj(z) annihilates a particle +at the point z in the jth well, where z is the coordinate +along the longitudinal direction (long axis of the system). +m is the mass of the particles, U(z) is a possible external +potential (in this paper it will be set to zero), g1D con- +trols the interparticle interaction strength, and J is the +tunneling frequency between the two wells. In the classi- +cal field approximation we replace the field operators by +complex functions +ˆψj(z) → ψj(z) = eiφj(z)� +n1D + ρj(z) . +(A2) +Note that φj and ρj are the phase and density variables +for each well rather than their antisymmetric versions +which are used extensively in the main text. +Let us start by manipulating the kinetic energy term +− +� +j=1,2 +� ∞ +−∞ +dz ℏ2 +2m +ˆψ† +j(z)∂2 ˆψj(z) +∂z2 +(A3) += +� ∞ +−∞ +dz +� +j=1,2 +ℏ2 +2m +� � ∂ +∂z e−iφj(z)� +n1D + ρj(z) +� +× +� ∂ +∂z e+iφj(z)� +n1D + ρj(z) +� � += +� ∞ +−∞ +dz +� +j=1,2 +ℏ2 +2m +� +− i∂φj +∂z +ˆψ† +j + +e−iφj ∂ρj +∂z +2√n1D + ρj +� +× +� +i∂φj +∂z +ˆψj + +eiφj ∂ρj +∂z +2√n1D + ρj +� += +� ∞ +−∞ +dz +� +j=1,2 +ℏ2 +2m +� +ˆψ† +j ˆψj +�∂φj +∂z +�2 ++ +( ∂ρj +∂z )2 +4(n1D + ρj) ++ i +∂ρj +∂z +∂φj +∂z +2√n1D + ρj +[ ˆψje−iφj − ˆψ† +jeiφj] +� += +� ∞ +−∞ +dz +� +j=1,2 +ℏ2 +2m +� +ˆψ† +j ˆψj +�∂φj +∂z +�2 ++ +( ∂ρj +∂z )2 +4(n1D + ρj) +� +≈ +� ∞ +−∞ +dz ℏ2 +2m +� +n1D +2 +��∂φs +∂z +�2 ++ +�∂φa +∂z +�2� ++ +1 +2n1D +��∂ρs +∂z +�2 ++ +�∂ρa +∂z +�2� � +(A4) +where +φa = φ1 − φ2, +φs = φ1 + φ2 +(A5) +ρa = ρ1 − ρ2 +2 +, +ρs = ρ1 + ρ2 +2 +, +(A6) +and we assume that n1D ≫ ρj. Next we consider the +interactions +� +j=1,2 +g1D +2 ψ† +jψ† +jψjψj = +� +j=1,2 +g1D +2 [n1D + ρj(z)]2 += +� +j=1,2 +� +g1Dn2 +1D +2 ++ g1Dρ2 +j +2 ++ g1Dn1Dρj +� +=g1Dn2 +1D + g1D(ρ2 +s + ρ2 +a) + 2g1Dn1Dρs . +(A7) +Finally, we consider the tunneling term +−ℏJ +� +ψ† +1(z)ψ2(z) + ψ† +2(z)ψ1(z) +� +(A8) += − ℏJ +� +(e−i(φ1−φ2) + e−i(φ2−φ1))√n1D + ρ1 +√n1D + ρ2 +� += − 2ℏJ cos(φa)√n1D + ρ1 +√n1D + ρ2 += − 2ℏJ cos(φa) +� +n2 +1D + 2n1Dρs + ρ2s − ρ2a +≈ − 2ℏJ cos(φa)(n1D + ρs) ≈ −2ℏn1DJ cos(φa) . +(A9) + +20 +At very low temperatures the symmetric and antisym- +metric components decouple and hence can be treated +separately. The lower energy terms are the antisymmet- +ric ones and we obtain the following Hamiltonian +HSG+ = +� ∞ +−∞ +dz +� +g1D ρa(z)2 + ℏ2n1D +4m +�∂φa +∂z +�2 ++ +ℏ2 +4mn1D +�∂ρa +∂z +�2 � +− +� ∞ +−∞ +dz 2ℏJn1D cos [φa(z)] . +(A10) +When the higher wavelength ρ modes are suppressed this +reduces to the sine-Gordon model +HSG = +� ∞ +−∞ +dz +� +g1D ρa(z)2 + ℏ2n1D +4m +�∂φa +∂z +�2 +− 2ℏJ n1D cos [φa(z)] +� +. +(A11) +Eq. (A11) is the finally obtained SG Hamiltonian HSG +which is the low energy description of two cigar shaped +tunnel-coupled quasicondensates [50, 74]. +Appendix B: Derivation of the Tomonaga-Luttinger +(TL) Hamiltonian in Fourier space +In this appendix we derive the Fourier space version +of the Tomonaga-Luttinger (TL) Hamiltonian. Starting +from Eq. (25), and applying the discrete Fourier decom- +positions given in Eq. (26) and Eq. (27), we have +HTL+(ra) = +� ∞ +−∞ +dz +g1D +NL + 1 +� +� +NL/2 +� +k=−NL/2 +ϱkei 2πkr +NL+1 +� +� × +� +� +NL/2 +� +l=−NL/2 +ϱlei 2πlr +NL+1 +� +� ++ +� ∞ +−∞ +dz +ℏ2n1D +4ma2(NL + 1) +∂ +∂r +� +� +NL/2 +� +k=−NL/2 +ϕkei 2πkr +NL+1 +� +� × ∂ +∂r +� +� +NL/2 +� +l=−NL/2 +ϕlei 2πlr +NL+1 +� +� ++ +� ∞ +−∞ +dz +ℏ2 +4mn1Da2(NL + 1) +∂ +∂r +� +� +NL/2 +� +k=−NL/2 +ϱkei 2πkr +NL+1 +� +� × ∂ +∂r +� +� +NL/2 +� +l=−NL/2 +ϱlei 2πlr +NL+1 +� +� += a +NL/2 +� +r=−NL/2 +NL/2 +� +k=−NL/2 +NL/2 +� +l=−NL/2 +� +�g1Dϱkϱlei 2π(k+l)r +NL+1 +NL + 1 +� +� +− a +NL/2 +� +r=−NL/2 +NL/2 +� +k=−NL/2 +NL/2 +� +l=−NL/2 +ℏ2n1D +4ma2(NL + 1) × +� +2π +NL + 1 +�2 +klϕkϕlei 2π(k+l)r +NL+1 +− a +NL/2 +� +r=−NL/2 +NL/2 +� +k=−NL/2 +NL/2 +� +l=−NL/2 +ℏ2 +4mn1Da2(NL + 1) × +� +2π +NL + 1 +�2 +klϱkϱlei 2π(k+l)r +NL+1 +(B1) +where we have split the z coordinate into NL + 1 grid +points separated by distance a so that z = r a where r +in an integer lying in the range specified by Eq. (11). +Using the fact that NLa = L, and applying the identity +�NL/2 +r=−NL/2 ei 2π(k+l)r +NL+1 += (NL + 1)δk,−l we obtain +HTL+ ≈a +� +k +� +l +g1Dϱkϱlδk,−l +− a +� +k +� +l +�ℏ2n1Dπ2 +mL2 +� +klϕkϕlδk,−l +− a +� +k +� +l +� +ℏ2π2 +mn1DL2 +� +klϱkϱlδk,−l +(B2) +where in the second term we have also replaced a2(NL + +1)2 by L2 which holds when NL ≫ 1. The limits of the +summation in Eq. (B2) has been omitted for the sake of + +21 +brevity. We therefore find +HTL+ ≈ +� +k +� +ag1Dϱkϱ−k+aℏ2n1Dπ2k2 +mL2 +ϕkϕ−k ++ aℏ2π2k2 +mn1DL2 ϱkϱ−k +� += +� +k +� +ag1D|ϱk|2+aℏ2n1Dπ2k2 +mL2 +|ϕk|2 ++ aℏ2π2k2 +mn1DL2 |ϱk|2 +� +(B3) +where we used the property of real fields that +ϕ−k = ϕ⋆ +k, +and +ϱ−k = ϱ⋆ +k . +(B4) +Hence the Hamiltonian takes the form given in Eq. (28) +of the main text. +Appendix C: Bench marking of the numerical +method +The results given in this paper rely on numerically +evolving the equations of motion over time for various +models [e.g. for the full SG+ model the equations of mo- +tion are given in Eq. (22)], which we accomplish using +the Julia package DifferentialEquations.jl [114]. This im- +plements a Runge-Kutta solver with a user-defined time +step. +As a measure of the accuracy of our numerical +method we use the deviation of the Hamiltonian from +its initial value. Since the Hamiltonian should be a con- +stant of motion this gives an indication of the size of the +numerical errors. +In Figures 12 and 13 we plot the relative error in the +SG+ Hamiltonian given in Eq. (18) for different time +and spatial resolutions. More precisely, Figure 12 shows +the effect of varying the time step d˜t, whereas Figure 13 +shows the effect of varying the number of grid points NL +which sets the spatial step d˜z. In both cases we have +evolved the system for a total elapsed time of ˜t = 1000 +which corresponds to the longest times we use in this +paper (for the calculation of the long-term distribution +shown in Figure 9), and also taken an ensemble average +over 100 different trajectories similar to those in Figure 5. +Furthermore, we also performed a moving time average +of 30-time steps around ˜t = 1000 to average out the effect +of fast oscillations. +As expected, the relative error decreases as d˜t and d˜z +decrease. For all the calculations in this paper we chose +d˜t = 0.2 and NL = 50 because this keeps the relative +error below 10 % and does not significantly slow down +the simulations. +Figure 12. +The relative error in the SG+ Hamiltonian is +plotted here as a function of the time step d˜t. The definition +of the SG+ Hamiltonian is given in Eq. 18 and should be +a constant of the motion were it not for numerical errors. +The moving time average of relative error is evaluated after +propagating the equations of motion for a total elapsed time +of ˜t = 1000. All parameter values are the same as in Figure +5 including NL = 50. +Figure 13. +The relative error in the SG+ Hamiltonian is +plotted here as a function of the number of lattice points NL +on the numerical spatial lattice. Like in Figure 12, the Hamil- +tonian is evaluated after evolving the equations of motion for +a total elapsed time of ˜t = 1000. The moving time average +of the relative error fluctuates (at around 10 %) but does de- +crease as d˜z decreases (or NL increases). All other parameter +values are the same as in Figure 5 with d˜t = 0.2 +Appendix D: Caustic curve +In this appendix we use the exact solution for the mo- +tion of a pendulum to calculate the caustic curve plotted +as the solid black line in Figure 3. The caustic is in fact +the envelope of a whole family of trajectories. To begin, +we take the equations of motion for the SG model given in + +0.6 +0.5 +(0) + 9SH (1) +0.4 +0.3 +0.2 +0.1 +10-1 +100 +101 +dt0.16 +0.14 +0.12 +0.10 +0.08 +0.06 +0.04 +20 +40 +60 +80 +100 +NL22 +Eq. (22) and drop the second order derivative term pro- +portional to ϵ which couples the different pendula. Next, +we make the change of variables +˜t = At, +˜ρ = Bp, +˜φ = 2y +(D1) +where +A = 1 +2 +1 +√J Γ +, +B = 2 +� +J +Γ +(D2) +so the equations of motion simplify to +dy +dt = p +(D3) +dp +dt = −1 +2 sin 2y . +(D4) +These equations are Hamilton’s equations obtained from +a standard pendulum hamiltonian of the form +H(y, p) = p2 +2 + 1 +2 sin2 y . +(D5) +The equations of motion given in Eqns. (D3) and (D4) +have exact solutions in terms of the Jacobi elliptic func- +tions sn[u|m] and cn[u|m] [122]. For the case relevant +to us where the pendulum starts at angle y0, with zero +initial angular momentum, they are +y(t, y0) = arcsin{sin y0 sn[t + K(sin y0)| sin y0]}(D6) +p(t, y0) = sin(y0) cn[t + K(sin y0)| sin y0] +(D7) +where K(m) = +� π/2 +0 +dθ/ +� +1 − m2 sin2 θ is the complete +elliptic integral of the first kind [122] (we caution the +reader that some computer packages such as Mathematica +use the syntax K(m2) for this integral). +Caustics occur when trajectories are focused, in other +words they are the places where the trajectory does not +change (to first order) when the initial conditions are +varied. Thus, caustics in the momentum variable p oc- +cur when dp/dy0 = 0 since the initial condition here is +specified by y0. By differentiating Eq. (D7) an implicit +expression for the position of the caustics can be found +[123] +sn(u|m)dn(u|m) +�E(am(−t|m) |m) +cos(y0) ++ t cos(y0) +� +− cos(y0)cn(u|m) = 0 +(D8) +where u = t+K(sin y0), m = sin y0, E(u|m) is an elliptic +integral of the second kind, dn(u|m) is another Jacobi +elliptic function, and am(u|m) = arcsin[sin(φ)/m] is the +Jacobi amplitude [122]. Finding the roots y0 of Eq. (D8) +numerically at each value of the time gives pairs of values +(y0, t) that can then be put back into Eq. (D7) to yield +the black curve for the caustic shown in Figure 3. The +match to the numerics is very good. +Appendix E: Derivation of ergodic (“circus tent”) +probability distribution at long times +In this appendix we outline the derivation of an an- +alytic approximation to the probability distribution for +the number difference at long times, as shown in Figure +9. This derivation is based upon a calculation given in +Ref. 124 and assumes that the average behaviour of a con- +tinuous chain of coupled pendula (the mechanical system +that underlies the sine-Gordon model) can be described +by a suitably ‘ergodized’ single pendulum. +To keep the calculation general we use the pendulum +Hamiltonian in standard form as given in Eq. (D5). With +this hamiltonian we define a microcanonical probability +density in phase space: +dm(y, p; y0) = +δ[H(y, p) − H(y0, p)] +� � +dy dp δ[H(y, p) − H(y0, p)] +(E1) +where y0 is the initial angle of the pendulum which fixes +its total energy to be E = (1/2) sin2 y0 if the the initial +angular momentum is zero (this is the appropriate ini- +tial condition for the tunneling quench considered in this +paper where the initial number difference is taken to be +zero), and the denominator ensures that dm is normalized +to unity. A microcanonical distribution has equal prob- +ability to be anywhere on its energy shell (in this case +a closed curve in y, p phase space) and thus by adopt- +ing Eq. (E1) we are making an ergodic hypothesis. This +does not hold for a single pendulum starting at position +y0 since it will spend the most time at its turning points +y = ±y0, but when averaged over y0 and y (see below) +it gives a very good approximation at long times, as can +be seen in Figure 9. +The normalization integral can be evaluated exactly +by re-expressing the delta function using the relation +δ[g(x)] = � +i δ(x − xi)/|g′(xi)|, where xi are the roots +of g(x). In the present case this gives +δ[(p2 + sin2 y − sin2 y0)/2] =δ[p − p1] +|p1| ++ δ[p − p2] +|p2| +=2δ[p − p1] +|p1| +(E2) +where |p1| = |p2| = +� +sin2 y0 − sin2 y. In obtaining this +expression we have used the fact that for values of y +within the range accessed by the pendulum, there are +two values of p where the integral crosses the energy shell. +The integral over p is now trivial due to the delta func- +tion and the integral over y can be performed by putting +sin y = sin y0 sin ζ so that + +23 +2 +� y0 +−y0 +dy +|p(y, y0)| = 2 +� y0 +−y0 +dy +� +sin2 y0 − sin2 y += 2 +� π/2 +−π/2 +dζ +� +1 − sin2 y0 sin2 ζ += 4 +� π/2 +0 +dζ +� +1 − sin2 y0 sin2 ζ += 4K(sin y0) . +(E3) +Therefore, the normalized microcanonical probability +density can be written as +dm(y, p; y0) = +1 +4K(sin y0)δ[(p2 + sin2 y − sin2 y0)/2] += +1 +2K(sin y0)δ(p2 + sin2 y − sin2 y0) +(E4) +where we have used the property of delta functions that +δ(αx) = (1/α)δ(x). +The initial condition for our dynamics is such that the +number difference is well defined but the phase differ- +ence is completely undefined. We must therefore average +the microcanonical probability density over all y0. This +gives the phase space probability density relevant to J- +quenches as being +W(y, p) = 1 +π +� π/2 +−π/2 +dy0 dm(y, p; y0) +(E5) +where we employ the notation W to indicate that this is +a classical version of the Wigner function. The properties +of the delta function can once more be used to write +δ(p2 + sin2 y − sin2 y0) = +� +i +δ(y0 − y0i)θ(cos y − |p|) +2 +� +p2 + sin2 y +� +cos2 y − p2 +(E6) +where θ(x) is the Heaviside step function. The integral +over y0 can now be evaluated exactly to give +W(y, p) = 2 +4π +θ(cos y − |p|) +K( +� +p2 + sin2 y) +� +p2 + sin2 y +� +cos2 y − p2 . +(E7) +The final step is to integrate out the y coordinate to +obtain the probability distribution PCT(p) for p alone +PCT(p) = +� π/2 +−π/2 +dy W(y, p) , +(E8) +where ��CT” stands for circus tent. Although this integral +cannot be done analytically, it can be put in a form which +is convenient to evaluate numerically. +Denoting m = +sin y0 = +� +p2 + sin2 y, one finds that +PCT(˜ρ) = +1 +2πB +� 1 +˜ρ2/B2 +dm +K(m) +� +m(1 − m)(m − ˜ρ2/B2)(1 + ˜ρ2/B2 − m) +(E9) +where we have also converted back from angular momen- +tum p to number difference ˜ρ using Eq. (D1). This equa- +tion is given in the main text as Eq. (44) and is plotted +in Figure 9 where it is compared against the long-time +spatially and temporally averaged numerical data for the +various nonlinear models considered in this paper. +As +can be seen in Figure 9, PCT is characterized by a di- +verging (yet normalizable) peak at the center and then +relatively flat wings until it drops sharply to zero at the +edges. In Ref. 124 it is shown that PCT(˜ρ) diverges log- +arithmically at the origin ˜ρ = 0 and also tends suddenly +to zero with logarithmic singularities at ˜ρ = ±B. Both +these non-thermal features can be attributed to the pres- +ence of caustics. +Appendix F: Pendulum at thermal equilibrium +In Figure 9 the long time probability distribution for +the number difference is compared against the ergodic +prediction derived in Appendix E, and also against the +thermal equilibrium prediction. In this Appendix we ex- +plain how to calculate the latter case. In order to make +the calculation tractable we make the assumption that +the SG+ model can be approximated by a thermal en- +semble of independent pendula. We also adopt the same +notation as Appendix E and hence work with a pendulum +Hamiltonian in the standard form H = (1/2)(p2+sin2 y). +This is related to the two mode Hamiltonian H2M = +Γ˜ρ2 − 2J cos φ by H = H2M/8J + 1/4. +We proceed in two steps: we first calculate the prob- +ability distribution PE(p) for the momentum variable p +(that here plays the role of the number difference) for a +fixed energy E. Secondly, we assume our system is at +thermal equilibrium with a bath at temperature T such +that the relative probability of any energy is given by the +Boltzmann factor exp[−E/T]. Thus the thermal proba- +bility distribution is +PT (p) = 1 +Z +� ∞ +0 +PE(p) e−E/T D(E) dE +(F1) +where Z is a normalizing factor (found numerically) and + +24 +D(E) is the density of states. +The probability distribution PE(p) at fixed E is pro- +portional to 1/ ˙p as this determines how long the pendu- +lum spends at each value of p. According to Hamilton’s +equation ˙p = −∂H/∂x = −(1/2) sin 2y, and using the +fact that sin y = +� +2E − p2, we find that this probability +distribution for a fixed value of E is +PE(p) = +N +(1/2) sin(2 arcsin +� +2E − p2) +, +(F2) +where N is a normalization factor given by the period +of the motion. +Two cases must be distinguished: for +E < 1/2 the energy is less than the separatrix and the +pendulum undergoes vibrational motion (also known as +librational motion in some literature). Conversely, when +E > 1/2 the energy is above the separatrix and the pen- +dulum undergoes rotational motion. +For motion below the separatrix we have |p| < pmax = +√ +2E. We must therefore supplement the expression for +PE(p) with the condition that it is zero if |p| > pmax and +this ensures that PE(p) is real. N is given in this case +by +N = +1 +2 K( +√ +2E) +(F3) +where, as in Appendix E, K is the complete elliptic inte- +gral of the first kind. +For motion above the separatrix we have +√ +2E − 1 < +|p| < +√ +2E and PE(p) is zero outside this range. N is +now given by +N = +√ +2E +4 K(1/ +√ +2E) +. +(F4) +To obtain the total thermal probability distribution +PT (p) given in Eq. (F1) we need the density of states +D(E) ≡ dn/dE, where n is the number of states be- +low energy E. According to the Bohr-Sommerfeld rule +n = S(E)/(2πℏ), where the action S(E) = +� +p dy is the +area in phase space enclosed by the energy contour E. +However, assuming that our Hamiltonian H is in units +ℏω then the 2πℏ factor is absorbed into the definitions of +p and y and we have D(E) = (d/dE) +� +p dy. Below the +separatrix we have +� +p(y)dy = 4 +� arcsin +√ +2E +0 +� +2E − sin2 y dy +(F5) +and putting 2E = sin2 y0 we find +D<(E) = d +dE +� +p(y)dy += 4 +� arcsin +√ +2E +0 +dy +� +sin2 y0 − sin2 y += 4K( +√ +2E) +(F6) +where the integral is performed in a similar fashion to +the one in Eq. (E3) and the subscript “<” indicates that +this is the expression valid below the separatrix. Above +the separatrix we find that the area enclosed in phase +space between two oppositely rotating states of the same +energy is +� +p(y)dy = 2 +� π/2 +−π/2 +� +2E − sin2 y dy +(F7) +and thus +D>(E) = d +dE +� +p(y)dy += 2 +� π/2 +−π/2 +dy +� +2E − sin2 y += +4 +√ +2E +K +� +1 +√ +2E +� +. +(F8) +Due to the fact that above the separatrix 2E > sin2 y we +no longer need to make the substitutions 2E = sin2 y0 +and sin y = sin y0 sin ζ, and the integral is straightfor- +ward. The subscript “>” indicates that this expression +holds above the separatrix. +We now have all the necessary ingredients to perform +the integral for PT (p) which we do numerically. +The +two contributions, one from below the separatrix and one +from above, are added together to get the total. Inter- +estingly, both density of states factors, Eqns. (F6) and +(F8), diverge at the separatrix such that the two con- +tributions individually display singular features but re- +markably these cancel out when the two parts are added +and result in the smooth gaussian curve plotted in Figure +9. +In order to compare the thermal distribution against +the quenched (followed by integrable SG evolution) dis- +tribution derived in Appendix E we need to choose a +temperature T for the thermal distribution PT . We do +this by matching the expectation value of the energy ⟨E⟩ +for both distributions. In the quenched case the initial +state corresponds to an ensemble of pendula with dif- +ferent starting angles y0 and zero kinetic energy. Each +starting angle in the range −π/2 < y0 ≤ π/2 is equally +probable in our J-quench. Therefore +⟨E⟩quench = 1 +π +� π/2 +−π/2 +1 +2 sin2 y0 dy0 = 1 +4 . +(F9) +To calculate ⟨E⟩ in the thermal case we compute +⟨E⟩T = 1 +ζ +� ∞ +0 +E e−E/T D(E) dE +(F10) +numerically for a large number of different values of +T, performing the integrals below and above the sep- +aratrix separately and adding the results. +Here ζ = +� ∞ +0 +e−E/T D(E) dE gives the normalization factor. We +then fit a curve to the results and find the value of T + +25 +that best matches the result given in Eq. (F9). We find +that T = 0.184 gives the best match. Putting back the +units this result is +kBT +8J ℏc/ξh += +kBT +16JℏK/π = 0.184 +(F11) +where c is the speed of sound and K is the Luttinger +parameter and J is the tunnel coupling rate between the +two wells. 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