Bi-directional universal dynamics in a spinor Bose gas close to a non-thermal fixed point
We numerically study the universal scaling dynamics of an isolated one-dimensional ferromagnetic spin-1 Bose gas. Preparing the system in a far-from-equilibrium initial state, simultaneous coarsening and refining is found to enable and characterize the approach to a non-thermal fixed point. A macroscopic length scale which scales in time according to , with , quantifies the coarsening of the size of spin textures. At the same time kink-like defects populating these textures undergo a refining process measured by a shrinking microscopic length scale , with . The combination of these scaling evolutions enables particle and energy conservation in the isolated system and constitutes a bi-directional transport in momentum space. The value of the coarsening exponent suggests the dynamics to belong to the universality class of diffusive coarsening of the one-dimensional XY-model. However, the universal momentum distribution function exhibiting non-linear transport marks the distinction between diffusive coarsening and the approach of a non-thermal fixed point in the isolated system considered here. This underlines the importance of the universal scaling function in classifying non-thermal fixed points. Present-day experiments with quantum gases are expected to have access to the predicted bi-directional scaling.
pacs:11.10.Wx 03.75.Lm 47.27.E-, 67.85.De
The dynamics of isolated quantum many-body systems quenched far out of equilibrium has been studied extensively in the recent past. Nonetheless, many questions remain open concerning, in particular, possible universal scaling on the way to equilibrium. For example, during prethermalization Gring et al. (2012); Langen et al. (2015); Aarts et al. (2000); Berges et al. (2004); Langen et al. (2016) quasi-stationary mode occupancies show trivial scaling in time, . Further universal phenomena include many-body localization Schreiber et al. (2015), critical and prethermal dynamics Braun et al. (2015); Nicklas et al. (2015); Navon et al. (2015); Eigen et al. (2018); Smale et al. (2018), decoherence and revivals Rauer et al. (2018), as well as wave- and superfluid turbulence Zakharov et al. (1992); Nazarenko (2011); Navon et al. (2016, 2018); Gauthier et al. (2018); Johnstone et al. (2018). Beyond these scenarios, if scaling occurs simultaneously in time and space, the evolution can become a type of renormalization-group flow, with time as the flow parameter, associated with the existence of a non-thermal fixed point Berges et al. (2008); Berges and Hoffmeister (2009); Scheppach et al. (2010); Prüfer et al. (2018); Erne et al. (2018). Such fixed points have been discussed and experimentally observed with Nowak et al. (2011, 2012); Schmidt et al. (2012); Schole et al. (2012); Karl et al. (2013, 2013); Karl and Gasenzer (2017); Erne et al. (2018) and without Berges et al. (2008); Berges and Hoffmeister (2009); Scheppach et al. (2010); Berges and Sexty (2011); Berges et al. (2014); Piñeiro Orioli et al. (2015); Berges (2016); Piñeiro Orioli and Berges (2018); Prüfer et al. (2018) reference to ordering patterns and kinetics, and topological defects, paving the way to a unifying description of universal dynamics.
Universal scaling in time and space of correlations of macroscopic observables in isolated systems is associated with the loss of information about the details of the initial condition and microscopic system properties. This universal scaling evolution is closely related to transport in momentum space associated with a few relevant symmetries only, and the corresponding conservation laws Piñeiro Orioli et al. (2015); Chantesana et al. (2018); Schmied et al. (2018); Mikheev et al. (2018). The ensuing universality renders such dynamics of strong interest across many different fields, including, besides cold gases, early-universe cosmology Kofman et al. (1994); Micha and Tkachev (2003); Berges et al. (2008); Gasenzer et al. (2012), and quark-gluon dynamics induced by nuclear collisions Berges et al. (2014, 2015).
In open classical systems coupled to a bath, universal scaling appears commonly in the context of dynamical critical phenomena Hohenberg and Halperin (1977); Janssen (1979), coarsening and phase-ordering kinetics Bray (1994), as well as glassy dynamics and ageing Calabrese and Gambassi (2005). For many-body quantum systems prethermal scaling Dalla Torre et al. (2013); Gambassi and Calabrese (2011); Sciolla and Biroli (2013); Smacchia et al. (2015); Maraga et al. (2015, 2016); Chiocchetta et al. (2015, 2016a, 2016b, 2017) as well as coarsening dynamics Damle et al. (1996); Mukerjee et al. (2007); Williamson and Blakie (2016a); Hofmann et al. (2014); Williamson and Blakie (2016b); Bourges and Blakie (2017) has been studied.
Coarsening is a specific type of universal scaling evolution, generically associated with the phase-ordering kinetics of a system coupled to a temperature bath, and exhibiting an ordering phase transition Bray (1994). Typically, the coarsening evolution following a quench into the ordered phase only involves a single characteristic length which fixes the scale of the correlations and evolves as a power law in time. Such a growing length scale can be associated with, e.g., the coarsening of magnetic domains in the ordered phase.
In general, universal scaling manifests itself in the evolution of correlations described, e.g., in momentum space, by a structure factor obeying the scaling form
where is a universal scaling function depending on a single variable only. The corresponding scaling exponents and define the evolution of the single characteristic length . The time scale denotes some reference time within the temporal scaling regime.
The concept of non-thermal fixed points generalizes such universal scaling dynamics to isolated systems far from equilibrium. Scaling dynamics near such a fixed point is associated with the transport of a conserved quantity, implying a behaviour (1) within a region of momenta in which the integral , with , remains invariant. Different conserved quantities can emerge in different momentum regimes, and thus the scaling dynamics violates single-length scaling. In this case the scaling evolution is expected to be characterized by multiple length scales with, in general, different scaling exponents.
Here we numerically demonstrate the violation of single-length scaling dynamics in one spatial dimension by studying the time evolution of an isolated spinor Bose gas after a sudden quench into the magnetically ordered phase. We find a bi-directional self-similar evolution of the structure factor characterized by the algebraic growth of an infrared (IR) scale associated with the conservation of local spin fluctuations as well as an algebraic decrease of a second scale connected to kinetic energy conservation in the ultraviolet (UV). The growth of is observed to be associated with the dilution of kink-like defects separating patches of approximately uniform spin orientation, while is set by the decreasing microscopic width of the defects, cf. Ref. Schmidt et al. (2012). Constraining the system to one spatial dimension, we find the IR scaling exponent . This value is considerably smaller than the standard exponent found in isolated systems for universal scaling transport towards the IR Schole et al. (2012); Piñeiro Orioli et al. (2015); Berges (2016); Karl and Gasenzer (2017); Chantesana et al. (2018); Prüfer et al. (2018); Schmied et al. (2018); Mikheev et al. (2018), associated with near-Gaussian fixed points Karl and Gasenzer (2017); Mikheev et al. (2018), and, in open systems in two and three dimensions, for diffusive coarsening of a non-conserved order parameter field Bray (1994). We emphasize that our findings are also in contrast to the case of a one-dimensional (1D) single-component gas where no scaling evolution is expected due to kinematic constraints on elastic scattering from energy and particle-number conservation and has been observed experimentally Erne et al. (2018).
Ii Spin-1 Bose gas in one spatial dimension
We consider a homogeneous one-dimensional spin-1 Bose gas described by the Hamiltonian Stamper-Kurn and Ueda (2013)
where is a three-component bosonic spinor field whose components account for the magnetic sublevels of the hyperfine manifold. is the quadratic Zeeman energy shift which is proportional to an external magnetic field along the -direction. It leads to an effective detuning of the components with respect to the component. We are working in a frame where a homogeneous linear Zeeman shift has been absorbed into the definition of the fields. Spin-independent contact interactions are described by the term , where is the total density. Spin-dependent interactions are characterized by the term , where is the spin density and is the spin in the fundamental representation. This term accounts. among others, for the redistribution of atoms between the three hyperfine levels Stamper-Kurn and Ueda (2013).
Spinor Bose gases can be realized in experiment in a well-controlled manner which makes them suitable for studying non-equilibrium phenomena Stenger et al. (1998); Ho (1998); Stamper-Kurn and Ueda (2013), see Refs. Sadler et al. (2006); Bookjans et al. (2011); Kawaguchi and Ueda (2012); Prüfer et al. (2018) for dynamics after a quench. Recently, universal scaling dynamics close to a non-thermal fixed point, with scaling exponent , has been observed experimentally in a ferromagnetic () spin-1 system in a near-1D geometry Prüfer et al. (2018). Theoretically, phase-ordering dynamics and scaling evolution has been studied in a ferromagnetic spin-1 Bose gas in 2D Williamson and Blakie (2016a, b, 2017); Symes and Blakie (2017) as well as in a ferromagnetic spin-1 Bose-Hubbard model in a 1D optical lattice Fujimoto et al. (2018a).
Apart from the trapping potential and a larger total density, we here perform numerical simulations in the parameter regime realized in the experiment Prüfer et al. (2018) on Rb in the hyperfine manifold. Assuming a constant homogeneous mean density , we can express the Hamiltonian (2) in terms of the dimensionless length , with spin healing length , time , with spin-changing collision time . The quadratic Zeeman shift is quantified by the dimensionless field strength , the field operators become , the density , the spin vector , and the dimensionless couplings read and . In the following, all quantities are expressed in the above units and the tilde will be suppressed.
In the ferromagnetic case (), and for a positive quadratic Zeeman energy , the equilibrium system exhibits two different phases separated by a quantum phase transition that breaks the spin symmetry of the ground state Kawaguchi and Ueda (2012). For the system, in its mean-field ground state, is in the polar phase and thus unmagnetized. On the opposite side of the transition, , the ground state is in the easy-plane ferromagnetic phase. Here, the non-conserved two-component order parameter is the transversal spin . Hence, the mean spin vector is lying in –-plane, bearing magnetization .
Iii Universal scaling dynamics
iii.1 Initial conditions and quench
We consider far-from-equilibrium dynamics after a quench, exerted on a homogeneous condensate in the polar phase, i.e. an initial state with , by means of a sudden change of the quadratic Zeeman shift to the parameter range . We compute the time evolution of observables using truncated Wigner simulations, starting each run with a field configuration for , with additional quantum noise added to the Bogoliubov modes Blakie et al. (2008); Polkovnikov (2010) of the polar condensate (see the appendix for details).
The quench induces transversal spin modes in the system to become unstable, leading to the formation of a spin-wave pattern during the early-time evolution after the quench. Non-linear interactions subsequently give rise to the formation of patches in the transversal spin. Within each patch, the phase angle of the complex order parameter is approximately constant in space (see Fig. 1b). At the same time, defects, represented by a dip in the amplitude and a corresponding phase jump, are traveling across the system at roughly the speed associated with the sound velocity of the spin degree of freedom (see solid lines in Fig. 1a). Spin patches in combination with phase jumps form spin textures whose size is given by the distance over which a phase winding occurs. The so-formed spin structure sets the stage for the subsequent ordering process. According to the evolution charts in Fig. 1 the average size of the textures appears to grow in time.
iii.2 Scaling evolution
For a quantitative analysis of the observed phase-ordering dynamics we consider averaged correlations of the order-parameter field. Since our system is translationally invariant on average, we evaluate these correlations in momentum space, by means of the structure factor
denoting the average over different runs. is formally obtained as the Fourier transform of with respect to .
Fig. 2a shows the time evolution of the structure factor for a quench to in the easy-plane ferromagnetic phase. The polar condensate at has no magnetization. At the population of the momentum modes of the structure factor within the instability regime fits the Bogoliubov prediction given by . Here, the growth rate of unstable momentum modes , with mode energy , is obtained as the imaginary part of the complex Bogoliubov mode energy Kawaguchi and Ueda (2012). In the course of the subsequent non-linear redistribution of the excitations the system is found to enter a spatio-temporal scaling regime where the structure factor evolves in a self-similar manner. During this period of the relaxation process, we observe three qualitatively different momentum regions which reflect the patterns seen in the single spatial realizations in Fig. 1. Below a characteristic momentum scale , the structure factor shows a plateau. Kink-like defects account for the power-law fall-off of the structure factor for momenta Bray (1994). The exponent depends on the dimensionality of the system and the defect structure. For kink-like defects () in one spatial dimension, the resulting exponent is close to the value which we extract by fitting the scaling form to the IR part of . For large momenta , the structure factor shows a steeper fall-off, before saturating at the level of ground-state fluctuations, , where denotes the lattice cutoff.
Fig. 2a indicates that the structure factor exhibits scaling according to Eq. (1) within a region of IR momenta below the UV end of the power-law fall-off, i.e., for . Taking the structure factor at time as a reference and performing a least-square fit of the data up to yields and . The errors are determined from the width of a Gaussian distribution used to fit the marginal-likelihood functions of both scaling exponents Piñeiro Orioli et al. (2015). These errors can become relatively large due to statistical uncertainties and systematic deviations caused by the limited scaling window.
Rescaling the structure factor in time by making use of the scaling form (1) yields the collapse onto a single curve below the momentum scale , as shown in Fig. 2b. The inset in Fig. 2b demonstrates that the local spin fluctuations are conserved in time (up to a relative error of ) within the IR scaling regime. Hence, we find, to a good approximation, that , with and . Using the scaling form (1) for the structure factor results in the scaling relation . The numerically extracted exponents are consistent with this scaling relation. As a consequence of the conserved local spin fluctuations we can describe the time evolution of the system for momenta by a single scaling exponent and thus a single characteristic IR length with . This macroscopic length scale corresponds to the size of the spin textures in the system.
iii.3 Violation of single-length scaling
In the UV range of momenta, however, the structure factor violates this single-length scaling and rather suggests a second characteristic length which shrinks in time. Fig. 2c shows that the rescaled structure factor collapses onto a single curve for momenta when choosing the scaling exponents and .
Hence, we find that the structure factor, in a range of momenta with strong spin-wave excitations, obeys the extended scaling form Chantesana et al. (2018)
with the scaling function being well approximated by
Here characterizes the large- fall-off. For the bi-directional scaling behavior shown in Fig. 2, the scaling function follows, to a good approximation, the form (5), with a single power-law exponent in between the IR and UV scales and , respectively. For the scaling function (5), the temporal scaling evolution of , implies that the exponents are related by . Moreover, imposing kinetic energy conservation in the UV scaling regime, i.e. within a corresponding UV momentum interval , , yields . Taking the additional conservation of local spin fluctuations in the IR, , one obtains the dynamical exponent , characterizing the dispersion , to be
Inserting the extracted parameters , and , we find . To cross-check this result we numerically determine the dispersion for which the kinetic energy shows the minimal deviation from being conserved within the UV scaling regime set by and with . This method yields consistent with the dynamical exponent directly calculated from the extracted scaling parameters.
Iv Discussion and conclusions
As the scaling dynamics of the spinor system takes place in the transversal spin, it is instructive to compare with similar behavior known for the 1D XY model. Considering an open system coupled to a heat bath and applying a temperature quench into the ordered phase leads to phase-ordering kinetics with a temporal scaling exponent Rutenberg and Bray (1995). At first sight, this appears to provide the universality classification for the self-similar dynamics seen in our system. However, the nature of the respective evolutions turns out to be qualitatively very different. Coarsening in the 1D XY model with non-conserved order parameter can be described as free phase diffusion of the order-parameter phase angle. As the topological charge is locally conserved in the system, the position-space correlation function at large distances is given by , where is the initial correlation length of the system. Thus, the characteristic IR length scale does not change in time. Instead, the scaling takes place in the UV giving rise to a broadening Gaussian spatial correlation function during the ordering dynamics, and thus to a sharpening Gaussian momentum-space structure factor Rutenberg and Bray (1995). In contrast, the ordering process in our isolated spinor gas is driven by non-linear dynamics of the spinor field, leading to a bi-directional transport of excitations in momentum space. This transport redistributes spin-wave excitations from an intermediate scale to both, smaller and larger wave numbers. Thereby, the correlation length grows in time as . Note that a similar behavior of the correlation length has been reported in the one-dimensional -state clock model for a moderate-sized in Ref. Andrenacci et al. (2006). We expect the coarsening dynamics described by this model to be closer to that of our system where the kink-like defects in the transversal spin are accompanied by phase jumps similar to the phase steps occurring in the -state clock model.
Self-similar evolution within a spatio-temporal scaling regime during the phase-ordering process of a non-equilibrium system is understood to generally occur as a transient phenomenon on the way to equilibrium. To study this transient nature we extract the time evolution of the characteristic momentum scales and by means of fitting the scaling form (5) to the structure factor for evolution times up to (see Fig. 6 for at time scales beyond ). We find that the IR scale shows scaling with for all times considered in our simulations (see Fig. 3a). To retain the IR scaling, energy has to be transported to the UV. In the case of the bi-directional scaling evolution the energy transported to the UV leads to the sharpening of kink-like defects. However, defects in the spin degree of freedom are expected to have a natural minimal width on the order of the spin healing length. As the UV scale approaches this length scale, i.e. as , we thus observe that the UV scaling exponent starts to deviate from (see Fig. 3b) causing the system to leave the regime of two-scale universal scaling dynamics. The deviation of the scaling exponent becomes clearly visible around . This behavior is accompanied by a build-up of a thermal tail in the range of momenta larger than , which instead stores the transported energy (cf. Fig. 6).
Although the system leaves the regime of two-scale universal scaling dynamics at , it remains close to the non-thermal fixed point as the IR scaling exponent is unaffected for evolution times up to . At a later point in time, which is presently beyond the reach of our simulations, we expect the rising mean kinetic energy in the thermal tail as well as the finite size of the system to induce the system to move away from the fixed point and towards final equilibrium.
In this work, we have numerically demonstrated universal self-similar dynamics in a one-dimensional ferromagnetic spin-1 Bose gas characterized by two separate time-evolving scales. While the IR scale increases as , with , the UV scale decreases as , with . Our results show that universal scaling evolution at a non-thermal fixed point is possible in a purely one-dimensional geometry, in contrast to standard arguments based on kinematic constraints prevailing for elastic collisions in 1D. The reported scaling is amenable to experiments with ultracold Bose gases while anticipated to be relevant also for very different systems in the relativistic realm.
Note added. After the completion of this paper a non-thermal fixed point associated with pair-annihilation of magnetic solitons has been reported for a one-dimensional antiferromagnetic spin-1 Bose gas Fujimoto et al. (2018b).
Acknowledgments. We thank J. Berges, P.B. Blakie, I. Chantesana, S. Erne, K. Geier, S. Heupts, M. Karl, P.G. Kevrekidis, P. Kunkel, S. Lannig, D. Linnemann, A.N. Mikheev, A. Piñeiro Orioli, J. Schmiedmayer, H. Strobel, and L. Williamson for discussions and collaborations on related topics. This work was supported by the Horizon-2020 framework programme of the European Union (FET-Proactive, AQuS, No. 640800, ERC Advanced Grant EntangleGen, Project-ID 694561), by Deutsche Forschungsgemeinschaft (SFB 1225 ISOQUANT), by Deutscher Akademischer Austauschdienst (No. 57381316), and by Heidelberg University (CQD, HGSFP).
Appendix A Initial state and post-quench dynamics
In this appendix we briefly discuss the representation of the ground states of our model in the polar and easy-plane phases and discuss the semi-classical numerical methods with which we have obtained our results presented in the main text.
a.1 Ground states in polar and easy-plane phase
In the case of ferromagnetic spin interactions () and for a positive quadratic Zeeman energy , the equilibrium system exhibits two different phases separated by a quantum phase transition that breaks the spin symmetry of the ground state Kawaguchi and Ueda (2012). For the system is in the polar phase where the mean-field ground state, given by the state vector
is unmagnetized. Here is a global phase distinguishing different realizations of the spontaneous symmetry breaking.
For the system is in the easy-plane ferromagnetic phase in which the mean-field ground state reads
Here denotes the angle with respect to the spin--axis. This ground state gives rise to the mean spin vector lying in the transversal spin plane, with magnetization .
a.2 Simulation methods
We consider out-of-equilibrium dynamics after a sudden quench, starting from a homogeneous condensate in the component, . We follow the time evolution by solving the coupled Gross-Pitaevskii equations (GPEs)
We consider experimentally relevant parameters for Rb in the hyperfine manifold, with . The initial condensate density is . The simulations are performed on a one-dimensional grid with grid points and periodic boundary conditions. The corresponding physical length is .
The initial state is given by a zero-temperature mean-field ground state in the polar phase, for , with additional quantum noise sampled from the positive definite Wigner distribution of the vacuum and set into the Bogoliubov modes of the polar condensate,
We again omit the tilde on the rescaled quantities , . The mode functions are complex Gaussian random variables with
which corresponds to adding an average occupation of half a particle in each mode . The Bogoliubov mode functions are given by
with mode energy .
Calculating observables using the truncated Wigner method requires averaging over many trajectories. We find that, in our one-dimensional geometry, a sufficient convergence of the observables is reached after averaging over trajectories.
Appendix B Universal scaling dynamics
In this appendix, we discuss, in more detail, the position-space correlation function in the scaling regime and demonstrate how the system departs from scaling during the late period of the evolution.
b.1 Spatial correlation function
The two different characteristic length scales that undergo universal scaling dynamics in the system can also be studied by means of the position-space correlation function which is calculated by applying a Fourier transform to the structure factor . Fig. 4 shows the position-space correlation function for distances within the temporal scaling regime. The shrinking of the characteristic UV length scale is found below distances . It is related to the quadratic part of the correlation function at short distances becoming steeper (see inset of Fig. 4). Note that the effect is small due to the slow scaling with . At larger distances the correlation function is given by
with time-evolving correlation length . The growth of the correlation length in time is associated with the decrease of the slope of the correlation function drawn in semi-logarithmic representation. This behavior is in contrast to coarsening dynamics of the 1D XY model where the Gaussian short-distance part grows in space while the slope of the exponential tail remains constant.
b.2 Scaling regime
Within the spatio-temporal scaling period, the scaling exponents are found to be independent of the reference time . Performing the least-square rescaling analysis for different reference times , within the constant time window with , we find that the scaling exponents settle to a constant value at which marks the onset of the scaling regime (see Fig. 5).
A constant fit to the extracted IR scaling exponents for yields and . For the UV scaling exponents we find and . The error is given by the standard deviation of all data points, which are not statistically independent. Making use of analyzing the scaling exponents for various reference times incorporates fluctuations of the scaling exponents caused by statistical errors in each reference spectrum thus leading to a more accurate determination of the universal exponents.
b.3 Departure from scaling
Scaling dynamics during the phase ordering process of a non-equilibrium system generically represents a transient process on the way to equilibration. Therefore we expect the system to leave the scaling regime at some time and relax back to its equilibrium state.
We are able to observe indications of this behavior when simulating the system times longer than presented in the main text. Fig. 6 shows the time evolution of the structure factor for on time scales up to . As shown in Fig. 3, the system leaves the two-scale self-similar regime at consistent with the data depicted in Fig. 6. For a thermal tail given by an approximate power law is formed in the UV. The temperature of the state is characterized by the slope of the power law. It slowly increases as time evolves up to . We expect the rising mean kinetic energy in the thermal tail to eventually cause the breakdown of the IR scaling. In consequence, the system is driven away from the non-thermal fixed point towards equilibrium. Note that the final equilibration process is beyond the time scales considered in our numerical simulations and depends on the particular IR and UV boundary conditions realized in the setup.
- Gring et al. (2012) M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Science 337, 1318 (2012).
- Langen et al. (2015) T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Science 348, 207 (2015).
- Aarts et al. (2000) G. Aarts, G. F. Bonini, and C. Wetterich, Phys. Rev. D 63, 025012 (2000).
- Berges et al. (2004) J. Berges, S. Borsanyi, and C. Wetterich, Phys. Rev. Lett. 93, 142002 (2004), arXiv:hep-ph/0403234 [hep-ph] .
- Langen et al. (2016) T. Langen, T. Gasenzer, and J. Schmiedmayer, J. Stat. Mech. 1606, 064009 (2016), arXiv:1603.09385 [cond-mat.quant-gas] .
- Schreiber et al. (2015) M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science 349, 842 (2015).
- Braun et al. (2015) S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera, M. del Rey, I. Bloch, J. Eisert, and U. Schneider, PNAS 112, 3641 (2015).
- Nicklas et al. (2015) E. Nicklas, M. Karl, M. Höfer, A. Johnson, W. Muessel, H. Strobel, J. Tomkovic, T. Gasenzer, and M. K. Oberthaler, Phys. Rev. Lett. 115, 245301 (2015), arXiv:1509.02173 [cond-mat.quant-gas] .
- Navon et al. (2015) N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Science 347, 167 (2015), arXiv:1410.8487 [cond-mat.quant-gas] .
- Eigen et al. (2018) C. Eigen, J. A. P. Glidden, R. Lopes, E. A. Cornell, R. P. Smith, and Z. Hadzibabic, Nature 563, 221 (2018), arXiv:1805.09802 [cond-mat.quant-gas] .
- Smale et al. (2018) S. Smale, P. He, B. A. Olsen, K. G. Jackson, H. Sharum, S. Trotzky, J. Marino, A. M. Rey, and J. H. Thywissen, ArXiv e-prints (2018), arXiv:1806.11044 [quant-ph] .
- Rauer et al. (2018) B. Rauer, S. Erne, T. Schweigler, F. Cataldini, M. Tajik, and J. Schmiedmayer, Science 360, 307 (2018).
- Zakharov et al. (1992) V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, Berlin, 1992).
- Nazarenko (2011) S. Nazarenko, Wave turbulence, Lecture Notes in Physics No. 825 (Springer, Heidelberg, 2011) pp. XVI, 279 S.
- Navon et al. (2016) N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Nature (London) 539, 72 (2016), arXiv:1609.01271 [cond-mat.quant-gas] .
- Navon et al. (2018) N. Navon, C. Eigen, J. Zhang, R. Lopes, A. L. Gaunt, K. Fujimoto, M. Tsubota, R. P. Smith, and Z. Hadzibabic, ArXiv e-prints (2018), arXiv:1807.07564 [cond-mat.quant-gas] .
- Gauthier et al. (2018) G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely, ArXiv e-prints (2018), arXiv:1801.06951 [cond-mat.quant-gas] .
- Johnstone et al. (2018) S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, and K. Helmerson, ArXiv e-prints (2018), arXiv:1801.06952v2 [cond-mat.quant-gas] .
- Berges et al. (2008) J. Berges, A. Rothkopf, and J. Schmidt, Phys. Rev. Lett. 101, 041603 (2008), arXiv:0803.0131 [hep-ph] .
- Berges and Hoffmeister (2009) J. Berges and G. Hoffmeister, Nucl. Phys. B813, 383 (2009), arXiv:0809.5208 [hep-th] .
- Scheppach et al. (2010) C. Scheppach, J. Berges, and T. Gasenzer, Phys. Rev. A 81, 033611 (2010), arXiv:0912.4183 [cond-mat.quant-gas] .
- Prüfer et al. (2018) M. Prüfer, P. Kunkel, H. Strobel, S. Lannig, D. Linnemann, C.-M. Schmied, J. Berges, T. Gasenzer, and M. K. Oberthaler, Nature 563, 217 (2018), arXiv:1805.11881 [cond-mat.quant-gas] .
- Erne et al. (2018) S. Erne, R. Bücker, T. Gasenzer, J. Berges, and J. Schmiedmayer, Nature 563, 225 (2018), arXiv:1805.12310 [cond-mat.quant-gas] .
- Nowak et al. (2011) B. Nowak, D. Sexty, and T. Gasenzer, Phys. Rev. B 84, 020506(R) (2011), arXiv:1012.4437v2 [cond-mat.quant-gas] .
- Nowak et al. (2012) B. Nowak, J. Schole, D. Sexty, and T. Gasenzer, Phys. Rev. A 85, 043627 (2012), arXiv:1111.6127 [cond-mat.quant-gas] .
- Schmidt et al. (2012) M. Schmidt, S. Erne, B. Nowak, D. Sexty, and T. Gasenzer, New J. Phys. 14, 075005 (2012), arXiv:1203.3651 [cond-mat.quant-gas] .
- Schole et al. (2012) J. Schole, B. Nowak, and T. Gasenzer, Phys. Rev. A 86, 013624 (2012), arXiv:1204.2487 [cond-mat.quant-gas] .
- Karl et al. (2013) M. Karl, B. Nowak, and T. Gasenzer, Sci. Rep. 3, 2394 (2013), 10.1038/srep02394, arXiv:1302.1122 [cond-mat.quant-gas] .
- Karl et al. (2013) M. Karl, B. Nowak, and T. Gasenzer, Phys. Rev. A 88, 063615 (2013), arXiv:1307.7368 [cond-mat.quant-gas] .
- Karl and Gasenzer (2017) M. Karl and T. Gasenzer, New J. Phys. 19, 093014 (2017), arXiv:1611.01163 [cond-mat.quant-gas] .
- Berges and Sexty (2011) J. Berges and D. Sexty, Phys. Rev. D 83, 085004 (2011), arXiv:1012.5944 [hep-ph] .
- Berges et al. (2014) J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Phys. Rev. D 89, 114007 (2014), arXiv:1311.3005 [hep-ph] .
- Piñeiro Orioli et al. (2015) A. Piñeiro Orioli, K. Boguslavski, and J. Berges, Phys. Rev. D 92, 025041 (2015), arXiv:1503.02498 [hep-ph] .
- Berges (2016) J. Berges, in Proc. Int. School on Strongly Interacting Quantum Systems Out of Equilibrium, Les Houches, edited by T. Giamarchi et al. (OUP, Oxford, 2016) arXiv:1503.02907 [hep-ph] .
- Piñeiro Orioli and Berges (2018) A. Piñeiro Orioli and J. Berges, (2018), arXiv:1810.12392 [cond-mat.quant-gas] .
- Chantesana et al. (2018) I. Chantesana, A. Piñeiro Orioli, and T. Gasenzer, (2018), arXiv:1801.09490 [cond-mat.quant-gas] .
- Schmied et al. (2018) C.-M. Schmied, A. N. Mikheev, and T. Gasenzer, arXiv:1807.07514 [cond-mat.quant-gas] (2018), arXiv:1807.07514 [cond-mat.quant-gas] .
- Mikheev et al. (2018) A. N. Mikheev, C.-M. Schmied, and T. Gasenzer, ArXiv e-prints (2018), arXiv:1807.10228 [cond-mat.quant-gas] .
- Kofman et al. (1994) L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994), arXiv:hep-th/9405187 .
- Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), hep-ph/0210202 .
- Gasenzer et al. (2012) T. Gasenzer, B. Nowak, and D. Sexty, Phys. Lett. B710, 500 (2012), arXiv:1108.0541 [hep-ph] .
- Berges et al. (2015) J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Phys. Rev. Lett. 114, 061601 (2015), arXiv:1408.1670 [hep-ph] .
- Hohenberg and Halperin (1977) P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
- Janssen (1979) H. Janssen, in Dynamical critical phenomena and related topics, Lecture Notes in Physics, vol. 104 (Springer, Heidelberg 1979, 1979) p. 26.
- Bray (1994) A. J. Bray, Adv. Phys. 43, 357 (1994).
- Calabrese and Gambassi (2005) P. Calabrese and A. Gambassi, J. Phys. A: Math. Gen. 38, R133 (2005).
- Dalla Torre et al. (2013) E. G. Dalla Torre, E. Demler, and A. Polkovnikov, Phys. Rev. Lett. 110, 090404 (2013).
- Gambassi and Calabrese (2011) A. Gambassi and P. Calabrese, Europhys. Lett. 95, 66007 (2011), arXiv:1012.5294 [cond-mat.stat-mech] .
- Sciolla and Biroli (2013) B. Sciolla and G. Biroli, Phys. Rev. B 88, 201110 (2013), arXiv:1211.2572 [cond-mat.stat-mech] .
- Smacchia et al. (2015) P. Smacchia, M. Knap, E. Demler, and A. Silva, Phys. Rev. B 91, 205136 (2015).
- Maraga et al. (2015) A. Maraga, A. Chiocchetta, A. Mitra, and A. Gambassi, Phys. Rev. E 92, 042151 (2015).
- Maraga et al. (2016) A. Maraga, P. Smacchia, and A. Silva, Phys. Rev. B 94, 245122 (2016).
- Chiocchetta et al. (2015) A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Phys. Rev. B 91, 220302 (2015).
- Chiocchetta et al. (2016a) A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Phys. Rev. B 94, 134311 (2016a).
- Chiocchetta et al. (2016b) A. Chiocchetta, A. Gambassi, S. Diehl, and J. Marino, Phys. Rev. B 94, 174301 (2016b).
- Chiocchetta et al. (2017) A. Chiocchetta, A. Gambassi, S. Diehl, and J. Marino, Phys. Rev. Lett. 118, 135701 (2017).
- Damle et al. (1996) K. Damle, S. N. Majumdar, and S. Sachdev, Phys. Rev. A 54, 5037 (1996).
- Mukerjee et al. (2007) S. Mukerjee, C. Xu, and J. E. Moore, Phys. Rev. B 76, 104519 (2007).
- Williamson and Blakie (2016a) L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116, 025301 (2016a).
- Hofmann et al. (2014) J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev. Lett. 113, 095702 (2014), arXiv:1403.1284 [cond-mat.quant-gas] .
- Williamson and Blakie (2016b) L. A. Williamson and P. B. Blakie, Phys. Rev. A 94, 023608 (2016b).
- Bourges and Blakie (2017) A. Bourges and P. B. Blakie, Phys. Rev. A 95, 023616 (2017).
- Stamper-Kurn and Ueda (2013) D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85, 1191 (2013).
- Stenger et al. (1998) J. Stenger, S. Inouye, D. Stamper-Kurn, H. Miesner, A. Chikkatur, and W. Ketterle, Nature 396, 345 (1998).
- Ho (1998) T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).
- Sadler et al. (2006) L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature 443, 312 (2006), cond-mat/0605351 .
- Bookjans et al. (2011) E. M. Bookjans, A. Vinit, and C. Raman, Phys. Rev. Lett. 107, 195306 (2011).
- Kawaguchi and Ueda (2012) Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012).
- Williamson and Blakie (2017) L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 119, 255301 (2017).
- Symes and Blakie (2017) L. M. Symes and P. B. Blakie, Phys. Rev. A 96, 013602 (2017).
- Fujimoto et al. (2018a) K. Fujimoto, R. Hamazaki, and M. Ueda, Phys. Rev. Lett. 120, 073002 (2018a).
- Blakie et al. (2008) P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Adv. Phys. 57, 363 (2008), arXiv:0809.1487v2 [cond-mat.stat-mech] .
- Polkovnikov (2010) A. Polkovnikov, Ann. Phys. 325, 1790 (2010), arXiv:0905.3384 [cond-mat.stat-mech] .
- Rutenberg and Bray (1995) A. D. Rutenberg and A. J. Bray, Phys. Rev. Lett. 74, 3836 (1995).
- Andrenacci et al. (2006) N. Andrenacci, F. Corberi, and E. Lippiello, Phys. Rev. E 74, 031111 (2006).
- Fujimoto et al. (2018b) K. Fujimoto, R. Hamazaki, and M. Ueda, arXiv:1812.03581 [cond-mat.quant-gas] (2018b).