Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent behavior
When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-Pitaevskii equation. Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with analytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscillations are absent. Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phenomenon.
pacs:05.60.Gg, 03.75.Lm, 03.75.Gg
The quantum-classical correspondence is well-established for single-particle quantum mechanics but is known to be problematic for some many-body quantum problems such as strongly correlated systems and even materials as simple as the antiferromagnet. A key macroscopic prediction of Bose-Einstein condensates (BECs) is the bright soliton, appearing as a localized robust ground state “lump” for attractive BECs. Based on the ubiquity of semiclassical limits for non-interacting and weakly interacting bosons, such as lasers and BECs, one might expect a well-defined emergent macroscopic classical behavior generically from such systems. To date, most aspects of matter-wave bright soliton experiments KhaykovichEtAl2002 (); StreckerEtAl2002 (); CornishEtAl2006 (); EiermannEtAl2004 (); MarchantEtAl2013 (); MedleyEtAl2014 (); NguyenEtAl2014 (); EverittEtAl2015 (); MarchantEtAl2016 (); LepoutreEtAl2016 (); McDonaldEtAl2014 () seem to be explained on the semiclassical mean-field level via the Gross-Pitaevskii equation (GPE): thus they display quantum behavior on a single-particle level matching classical wave experiments such as nonlinear photonic crystals sukhorukov1 () and spin-waves in ferromagnetic films kalinikos1998 (); kalinikos2000 (). This statement is supported by the fact that quantum-quantum bright solitons CarterEtAl1987 (); LaiHaus1989 (); CastinHerzog2001 (); CarrBrand2004 (); CalabreseCaux2007b (); SachaEtAl2009 (); MuthFleischhauer2010 (); BieniasEtAl2011 () — matter-wave bright solitons that display quantum behavior beyond the single-particle mean-field level — for many practical purposes show mean-field behavior predicted by the GPE emerging already for particle numbers as low as MazetsKurizki2006 (). So far, beyond-mean field effects only seem to play a role if two or more distinct bright solitons are involved: two matter-wave quantum bright solitons can behave quite differently from matter-wave mean-field bright solitons. Only the latter necessarily have a well-defined relative phase BillamWeiss2014 (). Both the limit of well-defined phase NguyenEtAl2014 () and the limit involving a superposition of many phases HoldawayEtAl2014 (); SakmannKasevich2016 () are experimentally relevant for matter-wave bright solitons NguyenEtAl2014 (); SakmannKasevich2016 (). In this Letter we show that truly quantum many-body effects are responsible for the dynamics of a single quantum-quantum bright soliton, a smoking-gun signal for quantum emergence in BEC experiments.
For far-from equilibrium dynamics of beyond-ground state quantum bright solitons, we are only at the beginning of a journey similar to the case of quantum dark solitons. That scientific voyage required multiple lines of investigations GirardeauWright2000 (); MishmashCarr2009 (); DelandeSacha2014 (); KronkeSchmelcher2014 (); KarpiukEtAl2015 () to arrive at the state-of-the art explanation that atom losses are necessary to obtain mean-field properties from many-body quantum solutions SyrwidSacha2015 (). Dark solitons were also realized experimentally in BECs BurgerEtAl1999 () and have been further explored in detail over the years in comparison to such predictions, e.g. weller2008 (). In contrast, bright solitons to-date lack for instance a phase coherence measurement, let alone the kind of far-from-equilibrium dynamics we are predicting here. Thus we focus on a quantum bright soliton experiment easily accessible in current platforms. Specifically, one first prepares a single ground-state bright soliton and then rapidly changes the interaction, an “interaction quench” via a Feshbach resonance, a well-established experimental technique. For one-dimensional Bose gases recent work related to quenches includes positive-to-negative quenches MuthFleischhauer2010 (); TschischikHaque2015 (), and zero-to-positive quenches ZillEtAl2014 (). Quenches involving dark-bright solitons SolnyshkovEtAl2016 (); LiuEtAl2016 (), quenched dynamics of two-dimensional solitary waves ChenEtAl2013 (), and breathers in discrete nonlinear Schrödinger equations JohanssonAubry2000 (); KevrekidisEtAl2000 (); LahiriEtAl2000 () were also investigated, as well as the breathing motion after a quench of the strength of a harmonic trap BarbieroSalasnich2014 ().
For attractive BECs, there are very specific mean-field predictions CarrCastin2002 (): in particular, for an interaction quench by a factor of four there are exact analytical mean-field results available that predict robust perfectly oscillatory behavior for all times SatsumaYajima1974 (). However, how quantum bright solitons would behave in such a situation is an open question which we address in the current Letter. One GPE interpretation of a higher order soliton is bound bright solitons, here , a kind of diatomic solitonic molecule in a nonlinear vibrational mode. One might therefore expect quantum fluctuations to cause the two solitons to unbind via e.g quantum tunneling out of a many-body potential, resulting in two equal-sized solitons moving away from each other StreltsovEtAl2008 (). This is not at all what we find, and is inconsistent with exact results for the center-of-mass wave function CosmeEtAl2016b (). Moreover, our beyond-mean-field results are distinct from the GPE failing for strongly correlated systems like Mott insulators FisherEtAl1989 (); JakschEtAl1998 (); GreinerEtAl2002 (); as well as from many-body systems on short timescales with differences disappearing for typical experimental parameters and large BECs GertjerenkenWeiss2013 (). We will show that an interaction quench leaves a large soliton core with small emissions of single particles. Experimentally these dynamics will appear as a “fizzled” higher order bright soliton, a stationary soliton core with a small thermal cloud. Thus we establish a new kind of quantum macroscopicity in weakly interacting bosonic systems.
The mean-field approach via the GPE is a powerful approximation which provides physical insight into weakly interacting ultracold atoms. In a quasi-one-dimensional wave guide carr2000e (); KhaykovichEtAl2002 (); StreckerEtAl2002 (); CornishEtAl2006 (); EiermannEtAl2004 (); MedleyEtAl2014 (); NguyenEtAl2014 (); EverittEtAl2015 (); MarchantEtAl2013 (); MarchantEtAl2016 (); LepoutreEtAl2016 (); McDonaldEtAl2014 () the GPE reads
where is a complex wave function normalized to unity and is the number of atoms of mass . The attractive interaction
is proportional to the s-wave scattering length and the perpendicular angular trapping-frequency, Olshanii1998 (). Some GPE predictions for repulsive BECs even become exact LiebEtAl2000 (); ErdosEtAl2007 () in the mean-field limit
While quantum bright solitons in their internal ground state in addition have a center-of-mass wavefunction (see Refs. GertjerenkenWeiss2012 (); DelandeEtAl2013 () and references therein), for measurements both many-body quantum physics CalogeroDegasperis1975 (); CastinHerzog2001 () and the GPE PethickSmith2008 () predict bright solitons localized at with a single-particle density profile of form
where the soliton length and the related soliton time
remain constant when approaching the mean-field limit (1).
In this Letter we use an interaction quench
After an interaction quench by a factor of , the GPE yields the analytical result (SatsumaYajima1974, , p 300)
Within the text book derivation PethickSmith2008 () of the GPE, the many-body wave function corresponding to the GPE is a Hartree-product state
The mean-square difference between the position of two particles
a measure that is distinct from and independent of the expansion of the center-of-mass wave function, can thus be calculated form the above analytical result to obtain
While the mean-field prediction thus is a perfectly periodic function of period , the question is what we expect to find on the many-body quantum level .
where denotes the position of particle of mass . The ground-state energy is given by
The energy eigenstates of excited states can be written as
corresponding to the intuitive interpretation of solitonlets — solitons that contain a fraction of the total number of particles — of size () and their individual center-of-mass kinetic energy. Equation (7) is valid if the system size is large compared to even a two-particle soliton — this can be included by adding a diverging system size to the mean-field limit (1) to get Weiss2016 (); HerzogEtAl2014 ()
Reaching such a limit is a difficult numerical problem AlcalaEtAl2016 (). However, by replacing the Hamiltonian (6) by the Bose-Hubbard model (BHM) used to model quantum bright solitons by e.g. GertjerenkenWeiss2012 (); BarbieroEtAl2014 (); GertjerenkenKevrekidis2015 (); DelandeEtAl2013 (), we introduce thermalization mechanisms present in real experiments such as a weak imperfectly harmonic trap, or a one-dimensional waveguide embedded in a 3D geometry; for the BHM thermalization is due specifically to a lattice, in our case in the limit of very weak discretization. The BHM takes the form
where () creates (annihilates) a particle on lattice site , quantifies the interaction energy of a pair of atoms and counts the number of atoms on lattice site . In order to use this in a way we can directly use the physical insight gained from the LLM (6), we choose for the hopping matrix element (cf. (GertjerenkenWeiss2012, , Eq. (17)))
such that both models have the same single-particle dispersion in the long wavelength limit , with the lattice constant. The interaction
While both the weak lattice introduced by Eq. (8)] and a weak harmonic trap MazetsSchmiedmayer2010 (); GringEtAl2012 () break the integrability of the LLM, we can still approximately describe eigenstates by Eq. (7). Furthermore, from a modeling point of view, by choosing the lattice we avoid the divergence of the energy fluctuations of the initial state immediately after the interaction quench, caused by delta functions squared, in . While in physics distributions with well-defined mean and diverging variance are well-known ZaburdaevEtAl2015 (), a more severe reason for avoiding the LLM limit is that this limit seems to be mathematically ill-defined – an initial wave function with the wrong boundary conditions at () has to be expressed in terms of eigenfunctions with the correct boundary conditions CastinHerzog2001 (). Summarizing, we note that these energy fluctuations are consistent with the LLM predicting the presence of quantum superpositions involving many solitonlets in the initial state, but inconsistent with simple pictures predicting two large solitonlets that either oscillate SatsumaYajima1974 () around each other or separate from each other StreltsovEtAl2008 () as the latter cannot happen rapidly CosmeEtAl2016b ().
We use time-evolving block decimation (TEBD) Vidal2004 () — a numerical method based on matrix product states Schollwock2005 (); WhiteFeiguin2004 () — to solve the BHM (8). In order to exclude both boundary effects and effects introduced by additional traps, we start with a very weak harmonic trap, such that opening it hardly introduces atom losses Castin2009 () and thus the analytical result (3) remains valid. In our simulations, we switch this trap off at the same time as we introduce the interaction quench.
If for quantum bright solitons indeed already show mean-field behavior MazetsKurizki2006 (), we should be able to see the mean-field oscillations depicted in Fig. 1 already for . Figures 2 and 3 show that the mean-field oscillations are absent from the many-body TEBD data for and up to 16, respectively. For the BHM (8), we note that the relative particle measurement of Eq. (5) can be rewritten in second quantized form appropriate to TEBD by replacing with .
Within the LLM, for relative distances large compared to the soliton length , the leading-order contributions to excited states consists of solitonlets of terms corresponding to individual solitonlets moving apart CastinHerzog2001 () In order to obtain a physical understanding on the time scales on which these solitonlets can move apart if they initially sit on top of each other, we recall the text book result for the variance of an initially Gaussian single-particle wave function Fluegge1990 (),
For the relative motion the mass has to be replaced by the relative mass . If initially localized to (much stronger localization leads to too high kinetic energies while much weaker localization leads to a too wide initial wave function) and for a relative mass independent of , the relative wave function will expand to a size larger than the initial wave function on time scales [cf. Eq. (2)]
The Hartree product states (4) are also ideal to calculate mean energies which in the mean-field limit (1) become identical to the exact many-body quantum results CastinHerzog2001 (). The kinetic energy prior to the interaction quench is , and the interaction energy . Immediately after the interaction quench, the kinetic energy remains unchanged and the interaction energy is increased to . In units of the new ground state energy we have a total average energy after the interaction quench of
where for and .
If ultracold attractive atoms are initially prepared in their ground state, by using the eigenstate thermalisation hypothesis PolkovnikovEtAl2011 (); EisertEtAl2015 () we conjecture that an interaction quench by a factor of will on short time scales lead to a final state consisting of a single bright soliton containing atoms, as given by thermodynamic predictions, and free atoms. In the mean-field limit (1), for bright solitons thermodynamic predictions read WeissGardinerGertjerenken2016 ()
i.e., one large soliton with reduced particle number and reduced size ; and single atoms which are not bound in molecules.
To suggest that this might indeed be what happens seems counterintuitive at best, since following “thermalization” according to the eigenstate thermalization hypothesis, all energetically accessible eigenfunctions will be involved PolkovnikovEtAl2011 (); EisertEtAl2015 (); violating both the Landau hypothesis (LandauLifshitz2002b, , very end), which at first glance seems to prevent co-existence of a large soliton and a free gas; argued against also by mean-field predictions SatsumaYajima1974 (); CarrCastin2002 () as well as thermodynamic predictions for ultracold atoms in contact with a heat bath HerzogEtAl2014 (); Weiss2016 (). However, the Landau hypothesis is based on assumptions that are not fulfilled for bright solitons WeissGardinerGertjerenken2016 () and thermally isolated ultracold atoms, arguably realised in state-of-the-art experiments with bright solitons KhaykovichEtAl2002 (); StreckerEtAl2002 (); CornishEtAl2006 (); EiermannEtAl2004 (); MarchantEtAl2013 (); MedleyEtAl2014 (); NguyenEtAl2014 (); EverittEtAl2015 (); MarchantEtAl2016 (); LepoutreEtAl2016 (); McDonaldEtAl2014 (), behave quite differently from those in contact with a heat bath WeissGardinerGertjerenken2016 (). Furthermore, contrary to rumours stating otherwise, one-dimensional Bose gases do thermalise, for example in the presence of a weak harmonic trap MazetsSchmiedmayer2010 (); GringEtAl2012 ().
As depicted in Fig. 4, we conjecture that after an interaction quench to more negative interactions, the system will relax towards the situation predicted in thermal equilibrium: the co-existence between one large soliton and a free gas WeissGardinerGertjerenken2016 (). The application of the eigenstate thermalisation hypothesis PolkovnikovEtAl2011 (); EisertEtAl2015 () is supported by the fact that single atoms will move the initial cloud much faster than larger solitonlets [Eq. (9)] that can continue to thermalize. Two macroscopic solitons sitting on top of each other would have to remain at the same position CosmeEtAl2016b (); a freely expending gas passes the convergence test of Ref. CosmeEtAl2016b () but energy conservation and Eq. (10) would require at least one soliton(let) to be present.
To conclude, we have combined evidence from three distinct models (GPE, LLM and BHM) to show that truly quantum emergent behaviour for attractive Bosons happens after an interaction quench to more attractive interactions. Combining the numerical evidence with general considerations based on the eigenstate thermalization hypothesis for larger particle numbers, we conjecture that the final many-body quantum state consists of one smaller bright soliton and lots of single atoms, thus yielding an ultimate example of a mean-field breakdown on time scales that remain experimentally relevant even in the mean-field limit (1). Our predictions are accessible to state-of-the-art experiments with thousands of atoms KhaykovichEtAl2002 (); StreckerEtAl2002 (); CornishEtAl2006 (); EiermannEtAl2004 (); MedleyEtAl2014 (); NguyenEtAl2014 (); EverittEtAl2015 (); MarchantEtAl2013 (); MarchantEtAl2016 (); LepoutreEtAl2016 (); McDonaldEtAl2014 (). Furthermore, the above conjecture offers an explanation as to why experiments that quasi-instantaneously switch from repulsive to attractive interactions (see for example Ref. MarchantEtAl2013 ()) while avoiding the “Bose-nova” collapse or modulational instability can nevertheless lead to one large matter-wave bright soliton (and a thermal cloud) being formed.
Acknowledgements.We thank T. P. Billam, J. Brand, J. Cosme, S. A. Gardiner, B. Gertjerenken, and M. L. Wall for discussions. We thank the Engineering and Physical Sciences Research Council UK for funding (Grant No. EP/L010844/). This material is based in part upon work supported by the US National Science Foundation under grant numbers PHY-1306638, PHY-1207881, and PHY-1520915, and the US Air Force Office of Scientific Research grant number FA9550-14-1-0287. L.D.C. thanks Durham University and C.W. the Colorado School of Mines for hosting visits to support this research. Data will be available online soon WeissCarr2016Data ().
- (1) L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon. Formation of a matter-wave bright soliton. Science, 296:1290, 2002.
- (2) K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet. Formation and propagation of matter-wave soliton trains. Nature (London), 417:150, 2002.
- (3) S. L. Cornish, S. T. Thompson, and C. E. Wieman. Formation of bright matter-wave solitons during the collapse of attractive Bose-Einstein condensates. Phys. Rev. Lett., 96:170401, 2006.
- (4) B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler. Bright Bose-Einstein gap solitons of atoms with repulsive interaction. Phys. Rev. Lett., 92:230401, 2004.
- (5) A. L. Marchant, T. P. Billam, T. P. Wiles, M. M. H. Yu, S. A. Gardiner, and S. L. Cornish. Controlled formation and reflection of a bright solitary matter-wave. Nat. Commun., 4:1865, 2013.
- (6) P. Medley, M. A. Minar, N. C. Cizek, D. Berryrieser, and M. A. Kasevich. Evaporative production of bright atomic solitons. Phys. Rev. Lett., 112:060401, 2014.
- (7) J. H. V. Nguyen, P. Dyke, D. Luo, B. A. Malomed, and R. G. Hulet. Collisions of matter-wave solitons. Nat. Phys., 10:918, 2014.
- (8) P. J. Everitt, M. A. Sooriyabandara, G. D. McDonald, K. S. Hardman, C. Quinlivan, M. Perumbil, P. Wigley, J. E. Debs, J. D. Close, C. C. N. Kuhn, and N. P. Robins. Observation of Breathers in an Attractive Bose Gas. ArXiv e-prints, 2015.
- (9) A. L. Marchant, T. P. Billam, M. M. H. Yu, A. Rakonjac, J. L. Helm, J. Polo, C. Weiss, S. A. Gardiner, and S. L. Cornish. Quantum reflection of bright solitary matter waves from a narrow attractive potential. Phys. Rev. A, 93:021604(R), 2016.
- (10) S Lepoutre, L Fouché, A Boissé, G Berthet, G Salomon, A Aspect, and T Bourdel. Production of strongly bound 39K bright solitons. ArXiv e-prints, 2016.
- (11) G. D. McDonald, C. C. N. Kuhn, K. S. Hardman, S. Bennetts, P. J. Everitt, P. A. Altin, J. E. Debs, J. D. Close, and N. P. Robins. Bright solitonic matter-wave interferometer. Phys. Rev. Lett., 113:013002, 2014.
- (12) A. A. Sukhorukov and Y. S. Kivshar. Spatial optical solitons in nonlinear photonic crystals. Phys. Rev. E, 65:036609, 2002.
- (13) B. A. Kalinikos, N. G. Kovshikov, and C. E. Patton. Self-generation of microwave magnetic envelope soliton trains in yttrium iron garnet thin films. Phys. Rev. Lett., 80:4301–4304, 1998.
- (14) B. A. Kalinikos, M. M. Scott, and C. E. Patton. Self-generation of fundamental dark solitons in magnetic films. Phys. Rev. Lett., 84:4697–4700, 2000.
- (15) S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby. Squeezing of quantum solitons. Phys. Rev. Lett., 58:1841, 1987.
- (16) Y. Lai and H. A. Haus. Quantum theory of solitons in optical fibers. ii. exact solution. Phys. Rev. A, 40:854, 1989.
- (17) Y. Castin and C. Herzog. Bose-Einstein condensates in symmetry breaking states. C. R. Acad. Sci. Paris, Ser. IV, 2:419, 2001.
- (18) L. D. Carr and J. Brand. Spontaneous soliton formation and modulational instability in Bose-Einstein condensates. Phys. Rev. Lett., 92:040401, 2004.
- (19) P. Calabrese and J.-S. Caux. Correlation functions of the one-dimensional attractive bose gas. Phys. Rev. Lett., 98:150403, 2007.
- (20) K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski. Anderson localization of solitons. Phys. Rev. Lett., 103:210402, 2009.
- (21) D. Muth and M. Fleischhauer. Dynamics of pair correlations in the attractive Lieb-Liniger gas. Phys. Rev. Lett., 105:150403, 2010.
- (22) P. Bienias, K. Pawlowski, M. Gajda, and K. Rzazewski. Statistical properties of one-dimensional attractive Bose gas. EPL (Europhys. Lett.), 96:10011, 2011.
- (23) I. E. Mazets and G. Kurizki. How different are multiatom quantum solitons from mean-field solitons? EPL (Europhys. Lett.), 76:196, 2006.
- (24) T. P. Billam and C. Weiss. Atomic solitons: These crashing waves. Nat. Phys., 10:902, 2014.
- (25) D. I. H. Holdaway, C. Weiss, and S. A. Gardiner. Phase-matching condition for enhanced entanglement of colliding indistinguishable quantum bright solitons in a harmonic trap. Phys. Rev. A, 89:013611, 2014.
- (26) K. Sakmann and M. Kasevich. Single-shot simulations of dynamic quantum many-body systems. Nat. Phys., 12:451, 2016.
- (27) M. D. Girardeau and E. M. Wright. Dark solitons in a one-dimensional condensate of hard core bosons. Phys. Rev. Lett., 84:5691, Jun 2000.
- (28) R. V. Mishmash and L. D. Carr. Quantum entangled dark solitons formed by ultracold atoms in optical lattices. Phys. Rev. Lett., 103:140403, 2009.
- (29) D. Delande and K. Sacha. Many-body matter-wave dark soliton. Phys. Rev. Lett., 112:040402, 2014.
- (30) S. Krönke and P. Schmelcher. Many-body processes in black and gray matter-wave solitons. Phys. Rev. A, 91:053614, 2015.
- (31) T. Karpiuk, T. Sowiński, M. Gajda, K. Rzazewski, and M. Brewczyk. Correspondence between dark solitons and the type II excitations of the Lieb-Liniger model. Phys. Rev. A, 91:013621, 2015.
- (32) A. Syrwid and K. Sacha. Lieb-liniger model: Emergence of dark solitons in the course of measurements of particle positions. Phys. Rev. A, 92:032110, 2015.
- (33) S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein. Dark solitons in Bose-Einstein condensates. Phys. Rev. Lett., 83:5198, 1999.
- (34) A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, M. K. Oberthaler, D. J. Frantzeskakis, G. Theocharis, and P. G. Kevrekidis. Experimental observation of oscillating and interacting matter wave dark solitons. Phys. Rev. Lett., 101:130401, 2008.
- (35) W. Tschischik and M. Haque. Repulsive-to-attractive interaction quenches of a one-dimensional bose gas in a harmonic trap. Phys. Rev. A, 91:053607, 2015.
- (36) J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer, and M. J. Davis. Relaxation dynamics of the Lieb-Liniger gas following an interaction quench: A coordinate Bethe-ansatz analysis. Phys. Rev. A, 91:023611, 2015.
- (37) D. D. Solnyshkov, A. V. Nalitov, and G. Malpuech. Kibble-Zurek Mechanism in Topologically Nontrivial Zigzag Chains of Polariton Micropillars. Phys. Rev. Lett., 116:046402, 2016.
- (38) I.-K. Liu, R. W. Pattinson, T. P. Billam, S. A. Gardiner, S. L. Cornish, T.-M. Huang, W.-W. Lin, S.-C. Gou, N. G. Parker, and N. P. Proukakis. Stochastic growth dynamics and composite defects in quenched immiscible binary condensates. Phys. Rev. A, 93:023628, 2016.
- (39) Q.-Y. Chen, P. G. Kevrekidis, and B. A. Malomed. Quenched dynamics of two-dimensional solitary waves and vortices in the Gross-Pitaevskii equation. J. Opt., 15:044012, 2013.
- (40) M. Johansson and S. Aubry. Growth and decay of discrete nonlinear Schrödinger breathers interacting with internal modes or standing-wave phonons. Phys. Rev. E, 61:5864, 2000.
- (41) P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop. Two-dimensional discrete breathers: Construction, stability, and bifurcations. Phys. Rev. E, 61:2006, 2000.
- (42) A. Lahiri, S. Panda, and T. K. Roy. Discrete Breathers: Exact Solutions in Piecewise Linear Models. Phys. Rev. Lett., 84:3570, 2000.
- (43) L. Barbiero and L. Salasnich. Phys. Rev. A, 89:063605, 2014.
- (44) L. D. Carr and Y. Castin. Dynamics of a matter-wave bright soliton in an expulsive potential. Phys. Rev. A, 66:063602, 2002.
- (45) J. Satsuma and N. Yajima. Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media. Prog. Theor. Phys. Suppl., 55:284, 1974.
- (46) A. I. Streltsov, O. E. Alon, and L. S. Cederbaum. Formation and Dynamics of Many-Boson Fragmented States in One-Dimensional Attractive Ultracold Gases. Phys. Rev. Lett., 100:130401, 2008.
- (47) J. G. Cosme, C. Weiss, and J. Brand. Center-of-mass motion as a sensitive convergence test for variational multi-mode quantum dynamics. Phys. Rev. A, 94:043603, 2016.
- (48) M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher. Boson localization and the superfluid-insulator transition. Phys. Rev. B, 40:546, 1989.
- (49) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonic atoms in optical lattices. Phys. Rev. Lett., 81:3108, 1998.
- (50) M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature (London), 415:6867, 2002.
- (51) B. Gertjerenken and C. Weiss. Beyond-mean-field behavior of large Bose-Einstein condensates in double-well potentials. Phys. Rev. A, 88:033608, 2013.
- (52) L. D. Carr, M. A. Leung, and W. P. Reinhardt. Dynamics of the Bose-Einstein condensate: Quasi-one-dimension and beyond. J. Phys. B: At. Mol. Opt. Phys., 33:3983–4001, 2000.
- (53) M. Olshanii. Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons. Phys. Rev. Lett., 81:938, 1998.
- (54) E. H. Lieb, R. Seiringer, and J. Yngvason. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A, 61:043602, 2000.
- (55) L. Erdős, B. Schlein, and H.-T. Yau. Rigorous derivation of the Gross-Pitaevskii equation. Phys. Rev. Lett., 98:040404, 2007.
- (56) B. Gertjerenken and C. Weiss. Nonlocal quantum superpositions of bright matter-wave solitons and dimers. J. Phys. B, 45:165301, 2012.
- (57) D. Delande, K. Sacha, M. Płodzień, S. K. Avazbaev, and J. Zakrzewski. Many-body Anderson localization in one-dimensional systems. New J. Phys., 15:045021, 2013.
- (58) F. Calogero and A. Degasperis. Comparison between the exact and Hartree solutions of a one-dimensional many-body problem. Phys. Rev. A, 11:265, 1975.
- (59) C. J. Pethick and H. Smith. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge, 2008.
- (60) E. H. Lieb and W. Liniger. Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev., 130:1605, 1963.
- (61) J. B. McGuire. Study of Exactly Soluble One-Dimensional N-Body Problems. J. Math. Phys., 5:622, 1964.
- (62) C. Weiss. Finite-temperature phase transition in a homogeneous one-dimensional gas of attractive bosons. ArXiv e-prints, 2016.
- (63) C. Herzog, M. Olshanii, and Y. Castin. Une transition liquide–gaz pour des bosons en interaction attractive à une dimension. Comptes Rendus Physique, 15:285, 2014.
- (64) D. A. Alcala, J. A. Glick, and L. D. Carr. Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein Condensates. ArXiv e-prints, 2016.
- (65) L. Barbiero, B. A. Malomed, and L. Salasnich. Quantum bright solitons in the Bose-Hubbard model with site-dependent repulsive interactions. Phys. Rev. A, 90:063611, 2014.
- (66) B. Gertjerenken and P.G. Kevrekidis. Effects of interactions on the generalized Hong-Ou-Mandel effect. Phys. Lett. A, 379:1737, 2015.
- (67) I E Mazets and J Schmiedmayer. Thermalization in a quasi-one-dimensional ultracold bosonic gas. New J. Phys., 12(5):055023, 2010.
- (68) M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer. Relaxation and Prethermalization in an Isolated Quantum System. Science, 337:1318, 2012.
- (69) V. Zaburdaev, S. Denisov, and J. Klafter. Lévy walks. Rev. Mod. Phys., 87:483–530, 2015.
- (70) G. Vidal. Efficient Simulation of One-Dimensional Quantum Many-Body Systems. Phys. Rev. Lett., 93:040502, 2004.
- (71) U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77:259, 2005.
- (72) S. R. White and A. E. Feiguin. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett., 93:076401, 2004.
- (73) Y. Castin. Internal structure of a quantum soliton and classical excitations due to trap opening. Eur. Phys. J. B, 68:317, 2009.
- (74) S. Flügge. Rechenmethoden der Quantentheorie. Springer, Berlin, 1990.
- (75) A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863, 2011.
- (76) J. Eisert, M. Friesdorf, and C. Gogolin. Quantum many-body systems out of equilibrium. Nat. Phys., 11:124, 2015.
- (77) C. Weiss, S. A. Gardiner, and B. Gertjerenken. Temperatures are not useful to characterise bright-soliton experiments for ultra-cold atoms. ArXiv e-prints, 2016.
- (78) E. M. Lifshitz and L. P. Pitaevskii. Landau and Lifshitz — Course of Theoretical Physics, Vol. 9: Statistical Physics, Part 1. Butterworth-Heinemann, Oxford, 2002.
- (79) C. Weiss and L. D. Carr. https://collections.durham.ac.uk/files/vt150j31m, http://dx.doi.org/10.15128/vt150j31m, 2016. “Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent behavior: Supporting data”.