Dissociation of one-dimensional matter-wave breathers due to quantum many-body effects

TL;DR

This paper predicts that quantum many-body effects cause the dissociation of one-dimensional matter-wave breathers, a phenomenon absent in mean-field models, and suggests it can be observed experimentally within current parameter ranges.

Contribution

It demonstrates the quantum many-body dissociation of matter-wave breathers using ab initio Bethe Ansatz dynamics, highlighting effects beyond mean-field theory.

Findings

Quantum many-body effects cause breather dissociation.

Mean-field models do not predict dissociation due to integrability.

Dissociation time is on the order of seconds for typical setups.

Abstract

We use the ab initio Bethe Ansatz dynamics to predict the dissociation of one-dimensional cold-atom breathers that are created by a quench from a fundamental soliton. We find that the dissociation is a robust quantum many-body effect, while in the mean-field (MF) limit the dissociation is forbidden by the integrability of the underlying nonlinear Schr\"{o}dinger equation. The analysis demonstrates the possibility to observe quantum many-body effects without leaving the MF range of experimental parameters. We find that the dissociation time is of the order of a few seconds for a typical atomic-soliton setting.

Equations

$$\hat{H}=-\frac{{\hslash }^2}{2m}\sum _{j=1}^N\frac{{\partial }^2}{\partial x_j^2}+g\sum _{j<j^{\prime }}\delta (x_j-x_{j^{\prime }}).$$

$${\left|⟨{\phi }_N^{(0)}|{\phi }_N⟩\right|}^2={\left(\frac{2\sqrt{|g{g}_0|}}{|g|+|g_0|}\right)}^{2(N-1)}={\left(\frac{4}{5}\right)}^{2(N-1)}$$

$$\frac{d{P}_{N_1,N-N_1}(v)}{dv}=(2-{\delta }_{N_1,N-N_1}){\left|⟨{\phi }_N^{(0)}|{\phi }_{N_1,N-N_1,v}⟩\right|}^2.$$

$$v_0=|g|/(2\hslash )\equiv a{\omega }_{\perp }$$

$$P_{N_1}\equiv {\int }_0^{\infty }\frac{d{P}_{N_1,N-N_1}(v)}{dv}dv,$$

$$P(v)=\sum _{N_1=1}^{\left[N/2\right]}\frac{d{P}_{N_1,N-N_1}(v)}{dv},$$

$$\Delta v\approx 0.39N^{0.54}{v}_0,$$

$$⟨v^2⟩={\int }_0^{\infty }{v}^{2}P(v)dv/{\int }_0^{\infty }P(v)dv$$

$$\sqrt{⟨v^2⟩}\approx 0.63N^{0.36}{v}_0,$$

$$\Delta v\approx 0.44\sqrt{N}{v}_0.$$

$$\delta {\tilde{x}}_{\Omega \to 0}\sim \delta v/\Omega .$$

$$\delta {\tilde{x}}_{\Omega \to \infty }\sim \sqrt{\hslash /(Nm\Omega )}.$$

$$\delta {\tilde{x}}_{\Omega \to 0}{|}_{\mu \sim \hslash \Omega }\sim \delta {\tilde{x}}_{\Omega \to \infty }{|}_{\mu \sim \hslash \Omega }\Rightarrow$$

$$\displaystyle\qquad\delta v\sim\sqrt{\frac{\hbar\Omega}{Nm}}\Big{|}_{\Omega\sim\frac{mg^{2}N^{2}}{\hbar^{3}}}\sim\frac{|g|}{\hbar}\sqrt{N}\,\,.$$

$$|\phi ⟩=\mathcal{N}\sum _{\mathcal{P}}A(\mathcal{P})\exp \left(i\sum _{j=1}^N{p}_{\mathcal{P}j}{x}_j/\hslash \right),$$

$$A(\mathcal{P})=\prod _{j<j^{\prime }}\left[1-\frac{img}{\hslash (p_{\mathcal{P}{j}^{\prime }}-p_{\mathcal{P}j})}\right]$$

$$p_j=i\frac{mg}{2\hslash }(N-2j+1)$$

$${\mathcal{N}}_N=\frac{1}{N\sqrt{L}}{\left(\frac{m|g|}{{\hslash }^2}\right)}^{(N-1)/2}$$

$$p_j=\{\begin{array}{ll}{P}_1+i\frac{mg}{2\hslash }(N_1-2j+1),& 1\le j\le N_1,\\ P_2+i\frac{mg}{2\hslash }(N+N_1-2j+1),& N_1<j\le N.\end{array}$$

$${\mathcal{N}}_{N_1{N}_{2}v}={\left(\frac{m}{2\pi N{N}_1{N}_2\hslash L}\right)}^{1/2}{\left(\frac{m|g|}{{\hslash }^2}\right)}^{(N-2)/2}$$

$$\times {\left(\frac{v^2+(N_1-N_2{)}^2{v}_0^2}{v^2+N^2{v}_0^2}\right)}^{1/2},$$

$$⟨{\phi }_N^{(0)}|{\phi }_{N_1{N}_{2}v}⟩={\mathcal{N}}_N^{(0)}{\mathcal{N}}_{N_1{N}_{2}v}N!{\hslash }^{N-1}L\sum _{\mathcal{P}}A(\mathcal{P})$$

$$\times \prod _{l=1}^{N-1}{\left\{\sum _{j=1}^l\left[i{p}_{\mathcal{P}j}-\frac{m{g}_0}{2\hslash }(N-2j+1)\right]\right\}}^{-1},$$

$$u(0,\tilde{t})=\exp \left(i\frac{5}{8}(1-{\tilde{v}}^2)\tilde{t}\right)\frac{(1+{\tilde{v}}^2)\cosh z\cos \varphi +2i({\tilde{v}}^2+1/4)\cosh z\sin \varphi +3/2\tilde{v}\sinh z\sin \varphi }{1/4+{\tilde{v}}^2+(1+{\tilde{v}}^2)\cosh 2z+3/4\cos 2\varphi },$$

$$\rho (x,t)=\mathrm{const}\underset{-\infty }{\overset{\infty }{\int }}dv\frac{d{P}_{N_1{N}_2}(v)}{dv}|u(x,\tilde{t}){|}^2.$$

$$\frac{d{P}_{N_1{N}_2}(v)}{dv}\approx \mathrm{const}\exp \left(-\frac{d_v}{N}{\left(\frac{v}{v_0}\right)}^2\right),$$

$$\Delta v\approx 0.44\sqrt{N}{v}_0$$

Full text

Under normal conditions, interacting quantum Bose gases do not readily exhibit signatures of their corpuscular nature, but rather follow the behavior predicted by mean-field (MF) theory. The observability of microscopic quantum effects involving a substantial fraction of the particles in a coherent macroscopic setting generally requires going beyond-MF, for example, at low density in 1D Kinoshita et al. (2004); Tolra et al. (2004) or high density in 3D. In 3D systems, the high-density Lee-Huang-Yang corrections, which are induced by quantum correlations, were realized experimentally using the Feshbach resonance Navon et al. (2011) and in the spectacular form of “quantum droplets” in dipolar Ferrier-Barbut et al. (2016); Schmitt et al. (2016); Chomaz et al. (2016) and isotropic Cabrera et al. (2017) bosonic gases., i.e., as self-trapped states stabilized against the collapse by the beyond-MF self-repulsion. This stabilization was predicted in Refs. Petrov (2015); Petrov and Astrakharchik (2016); Baillie et al. (2016). Quantum effects involving a macroscopic number of atoms in collapsing attractive 3D gases and colliding condensates were also observed Sackett et al. (1999); Gerton et al. (2000); Donley et al. (2001); Vogels et al. (2002); Chin et al. (2003) and analyzed Yurovsky (2002); Yukalov and Yukalova (2004) in the MF density range.

A generic opportunity to observe beyond-MF effects arises when a particular symmetry of the MF dynamics, which prohibits a certain effect, is broken at the microscopic level thus making observation of the effect possible. For instance, the scale invariance in the dynamics of a harmonically trapped 2D Bose gas nullifies the interaction-induced shift of the frequency of monopole excitations for all excitation amplitudes; however, this scale invariance is broken by the quantum many-body Hamiltonian, leading to a small shift, albeit discernible on a zero background Olshanii et al. (2010). In this context, the symmetry breaking by the secondary quantization may be considered as a manifestation of a general phenomenon known as the quantum anomaly Gupta and Rajeev (1993). In this Letter we develop a similar strategy for predicting beyond-MF effects in the one-dimensional (1D) self-attractive Bose gas in a MF range of parameters. The respective MF equation amounts to the nonlinear Schrödinger (NLS) equation, integrable by the inverse-scattering transform Ablowitz and Segur (1981). The NLS rigidly links the structure of a time-dependent solution to its initial form, with many features of the latter rendered identifiable in the former. In particular, a sudden increase of the strength of the attractive coupling constant by a factor of $4$, i.e. an interaction quench, converts a fundamental soliton into an exact superposition of two solitons with zero relative velocity, zero spatial separation, and with a mass ratio $3:1$ Zakharov and Shabat (1972); Satsuma and Yajima (1974); Carr and Castin (2002). The two superimposed solitons have different chemical potentials, hence the density oscillates as a result of interference. Such an exact superposition of fundamental solitons is identified as an NLS breather.

Further, quantum fluctuations in solitons have also been analyzed in terms of the exact Bethe-ansatz (BA) solution Berezin et al. (1964); McGuire (1964); Calogero and Degasperis (1975); Lai and Haus (1989a), the linearization approximation Lai and Haus (1989b), and the numerical positive-$P$ representation Carter et al. (1987); Drummond and Carter (1987). These effects have been observed in experiments Rosenbluh and Shelby (1991); Drummond et al. (1993); Corney et al. (2006, 2008), see also review Drummond and Chaturvedi (2016). In particular, in the quantum many-body theory, contrary to its MF counterpart, the center-of-mass (COM) position of a soliton is a quantum coordinate whose conjugate momentum is subject to quantum fluctuations Drummond and Man (1994); Vaughan et al. (2007); Castin (2009).

The MF breather generated by the quench does not split due to the absence of any relative velocity in the MF. We predict, however, that the spread of the relative velocity of the two solitons leads to dissociation, and thus reveals a many-body quantum effect. A different dissociation scenario was predicted in Streltsov et al. (2008).

In Ref. Weiss and Carr it is shown, using a Bose-Hubbard model, that higher-order solitons also break up due to many-body quantum effects. The fact that a nonintegrable lattice model, with thermalization of eigenstates, also predicts many-body quantum effects is relevant for comparison with results of the present work.

We consider $N$ atoms of mass $m$ moving in a waveguide with a transverse trapping frequency $\omega_{\perp}$. In the “deep 1D” approximation they can be considered as particles moving in the $x$ direction, with zero-range interactions of strength $g=2\hbar a\omega_{\perp}$ Olshanii (1998), where the $s$-wave scattering length $a$ can be tuned by an external magnetic field via the Feshbach resonance. The corresponding Lieb-Liniger Hamiltonian is Yurovsky et al. (2008)

[TABLE]

This problem has an exact BA solution Lieb and Liniger (1963); Berezin et al. (1964). Due to the translational invariance of the Hamiltonian (1), its eigenfunctions are delocalized, having a homogeneous density. For attractive interactions with $g<0$, there are also eigenstates in the form of one or several several strings – bound states of several particles, i.e., quantum solitons Berezin et al. (1964); McGuire (1964); Yang (1968). Although they remained a theoretical concept since they were introduced, very recently similar states — the Bethe strings — have been directly observed in an antiferromagnetic Heisenberg-Ising chain Wang et al. ). A superposition of strings with different velocities may remain localized for a finite time, so that it carries over into a MF multi-soliton (breather) in the limit of $N\rightarrow\infty$ Calogero and Degasperis (1975); Lai and Haus (1989b, a). Normalization factors for multi-string states were derived in Ref. Calabrese and Caux (2007).

We assume that, at $t<0$, the interaction strength was $g_{0}=g/4$, and the system contained a single-string state $\varphi_{N}^{(0)}$ with zero COM velocity. At $t=0$, the external magnetic field suddenly changes, switching the interaction strength to $g$, i.e., applying a $4$-fold quench to the system. The exact BA calculation, starting from the quenched state, makes it possible to directly compare the result in the quantum many-body system with its exactly known MF counterpart — the second-order breather, which is generated by the $4$-fold quench Satsuma and Yajima (1974). This is, essentially, the objective of the present work.

After the application of the quench, the many-body configuration will be a superposition of a single-string state $\varphi_{N}$, double-string states $\varphi_{N_{1},N-N_{1},v}$, where $v$ is the relative velocity of two strings composed of $N_{1}$ and $N-N_{1}$ atoms, and multi-string states. On the other hand, a fundamental quantum soliton is a superposition of the single-string states with different COM velocities. These states are mutually orthogonal due to the COM velocity conservation, therefore probabilities of quench-triggered transitions from the pre-quench fundamental-soliton state to multi-soliton ones will be the same as for the delocalized string states. The probabilities are calculated analytically using the exact BA solution sup . It is the basic technical result of the present work, which underlies the physical considerations. First, the probability to remain in the single-string state is

[TABLE]

For the double-string states the probabilities depend on the relative string velocity $v>0$ and the string composition,

[TABLE]

It is a sum of the probabilities corresponding to velocities $v$ and $-v$ for $N_{1}\neq N-N_{1}$, while for $N_{1}=N/2$ the states with $v$ and $-v$ are identical. Examples of the probabilities are displayed in Fig. 1, for $N=4$ and $N=20$. The natural velocity scale is

[TABLE]

Total probabilities of the transition to double-string states with fixed $N_{1}$,

[TABLE]

are presented in Fig. 2 making it obvious that the transition $N\rightarrow 3N/4+N/4$ features the largest probability, in agreement with the MF prediction. The cumulative probability of the transition to all double-string states, $\sum_{N_{1}=1}^{\left[N/2\right]}P_{N_{1}}$, exceeds $80\%$ for $N\geq 8$ (here, $\left[...\right]$ stand for the integer part).

Another similarity to the MF is seen in the fact that the quench-produced configuration, being a superposition of multi-string eigenstates with different energies, oscillates in time due to their interference, thus qualitatively resembling the breather. The binding energy of the multi-string solution is the sum of the constituting string energies, each one being $E_{N_{1}}=-N_{1}(N_{1}^{2}-1)mg^{2}/(24\hbar^{2})$ Berezin et al. (1964); McGuire (1964). In particular, the binding-energy difference between the $(N_{1},N-N_{1})$ and $(N_{1}-1,N-N_{1}+1)$ double-string states leads to beatings at frequency $[E_{N_{1}-1}+E_{N-N_{1}+1}-(E_{N_{1}}+E_{N-N_{1}})]/\hbar=N[N_{1}-(N+1)/2]mg^{2}/(4\hbar^{3})$, which tends to the the MF breather frequency, $mg^{2}N^{2}/(16\hbar^{3})$, at $N_{1}=3N/4\rightarrow\infty$.

Probability distributions for the relative velocity of the dissociation products, summed up over all double-string dissociation channels,

[TABLE]

is almost independent of $N$, see its plot as a function of $v/\sqrt{N}$ in Fig. 3.

The numerically calculated half-width at half-maximum (HWHM), $\Delta v$, of the velocity distribution, defined by $P(\Delta v)=P(0)/2$, can be fitted to the following formula, which is, naturally, close to the $\sqrt{N}$ dependence:

[TABLE]

see Fig. 4. The relative velocity can be measured also by its mean-square value,

[TABLE]

However, the numerically found root-mean-square (r.m.s.) velocity increases with $N$ only as

[TABLE]

according to the fit displayed in Fig. 4. The probability distribution (2) has slowly decaying tails for small $N$, in particular, $dP_{3N/4,N/4}(v)/dv\sim v^{-3N}$ at $v\rightarrow\infty$. The tails increase the r.m.s. velocity at small $N$ and, therefore, slower its gain with $N$. On the contrary, due to the normalization condition, the tails exhaust the width of the central part of the $v$ distribution at small $N$, boosting the HWHM growth with $N$. Then at large $N$, when the tail effects fade out, the r.m.s. velocity and HWHM should gain faster and slower, respectively, than at small $N$. These arguments suggest that both measures of the relative velocity variation assume the same asymptotic scaling at large $N$, which should be close to $\sqrt{N}$, according to Fig. 3. The eventual fit is displayed in Fig. 4:

[TABLE]

The following estimate confirms the $\sqrt{N}$ scaling for a typical relative velocity of the solitons, $\delta v$. Consider the system placed in an external harmonic-oscillator (HO) potential with frequency $\Omega$. Varying $\Omega$ from vanishingly small values towards very large ones, at each $\Omega$ one can apply the $g/4\rightarrow g$ quench to the respective ground state. The figure of merit to monitor is $\delta\tilde{x}$—the time-averaged distance, further symmetrized over permutations, between COMs of two groups of atoms, each containing the number of atoms $\sim N$. At small $\Omega$, the state obtained right after the quench is unaffected by the external confinement, hence the two solitons (strings) start their motion with the free-space relative velocity $\delta v$. Thus, the distance $\delta\tilde{x}$ will be dominated by the typical distance between the solitons placed in the HO potential, $\delta v/\Omega$, which diverges at small $\Omega$. This very long scale governs the estimate for $\delta\tilde{x}$, the other potentially relevant length scale, the average distance between two atoms inside the same soliton, which is on the order of the size of an individual soliton, $\sim\hbar^{2}/(m|g|N)$, does not diverge at $\Omega\rightarrow 0$. Thus,

[TABLE]

On the other hand, at large $\Omega$, the effect of the interatomic interactions vanishes and the estimate for $\delta\tilde{x}$ is determined by zero-point quantum fluctuations of the COM position of the cloud containing $\sim N$ particles:

[TABLE]

A crossover between the two regimes occurs when the interaction energy per particle (comparable to the chemical potential of the gas, $\mu$), estimated as $\sim\mu\sim mg^{2}N^{2}/\hbar^{2}$, becomes comparable to the HO quantum, $\hbar\Omega$. Indeed, when the former is dominated over by the latter, the interactions are irrelevant, and the system becomes an HO-confined ideal gas. At the crossover, the two above-mentioned estimates yield the same value. An estimate for $\delta v$ immediately follows:

[TABLE]

Indeed, this estimate is consistent with the fit (8).

The above results suggest that experimental observation of the variance in the relative velocity of the solitons due to quantum many-body effects may be possible. To demonstrate this, we consider $3\times 10^{3}$ 7Li atoms, in a waveguide with transverse trapping frequency $\omega_{\perp}=2\pi\times 254$ Hz. The initial state is a fundamental matter-wave soliton, existing at scattering length $a_{t<0}=-1.0\,a_{\text{Bohr}}$, which is quenched up to $a_{t>0}=-4\,a_{\text{Bohr}}$ 111For the present set of parameters, the number of atoms is only two times lower than the collapse critical number, $N_{\text{crit}}=7900$, hence deviations from the purely 1D behavior are expected. However, moderate 3D corrections do not destroy MF breathers Golde et al. . The resulting state constitutes an NLS breather with an aphelion density profile proportional to $\text{sech}^{2}(x/\ell_{\text{breather}})$ and width $\ell_{\text{breather}}=8\hbar^{2}/(mgN)=36$ $\mathrm{\mu}$m Zakharov and Shabat (1972); Satsuma and Yajima (1974). Assuming that the splitting of the breather into two solitons becomes apparent when the distance between their COMs, after evolution time $\tau$, $\Delta x=\Delta v\cdot\tau$, becomes comparable to the breather’s width $\ell_{\text{breather}}$, and using extrapolation (6) for the relative velocity of the solitons, we obtain $\tau\simeq 3$ s for the time necessary to certainly observe the splitting of the breather caused by the quantum dynamics.

The predicted dissociation time can be made even shorter at the expense of reducing the cloud population, assuming that the scattering length simultaneously increases so as to keep product $Na$ at a finite fraction of the collapse critical value, $Na\lesssim a_{\perp}$, $a_{\perp}\sim\sqrt{\hbar/(m\omega_{\perp})}$ being the size of the transverse vibrational ground state of the waveguide used. The microscopic velocity scale $v_{0}$, the separation velocity $\Delta v$, and the breather size $\ell_{\text{breather}}$ can be estimated as $v_{0}\lesssim\hbar/(ma_{\perp}N)$, $\Delta v\lesssim\hbar/(ma_{\perp}\sqrt{N})$, and $\ell_{\text{breather}}\gtrsim a_{\perp}$, respectively. Then the breather dissociation time diminishes as $\tau\sim\ell_{\text{breather}}/\Delta v\gtrsim(1/\omega_{\perp})\sqrt{N}$ with the decrease of the number of particles.

For the analysis of possibilities for the experimental implementation of the predictions reported here, it is important to estimate deviations of real-world settings from the idealized model Kivshar and Malomed (1989); Sakaguchi and Malomed (2004); Yanay et al. (2009). In this connection, it is essential to consider the departure from the one-dimensionality, as suggested, in particular, by the work aimed at experimental observation of the quantum violation of the scale-invariance-induced constancy of the monopole-mode frequency in the 2D Bose gas. In that case, weak dependence of the quantum state on the third, confined dimension tends to mask the quantum many-body effects Merloti et al. (2013). Nevertheless, experiments have clearly demonstrated that 3D experimental setups with appropriately designed transverse confinement can be efficiently used for the emulation of ideal one-dimensional quantum settings, and such emulations are stable against real-world disturbances. Relevant examples are the creation of the atomic Newton’s cradle with repulsive interactions Kinoshita et al. (2006), and the realization of the super-Tonks-Girardeau gas Haller et al. (2009). The latter example is especially relevant for the comparison with the present analysis, as it is also based on attractive interactions. Predictions of the MF counterpart of the Lieb-Liniger model, i.e., the Gross-Pitaevskii equation, which are also based on the one-dimensionality and integrability, are very well confirmed in numerous experiments with matter-wave solitons Strecker et al. (2002); Khaykovich et al. (2002); Marchant et al. (2013); Nguyen et al. (2014); Marchant et al. (2016). The well-known stability of the exact solution of the Lieb-Liniger model Lieb and Liniger (1963); Berezin et al. (1964) clearly means that the results may only be slightly perturbed by other distortions, such as external fluctuations and inhomogeneities.

As concerns the full 3D analysis, an example which makes it possible to explicitly compare the MF approximation and its many-body counterpart is offered by the problem of the stabilization of the gas of bosons with repulsive interactions, attracted to the center with potential $\sim-r^{-2}$. In that case, the MF predicts suppression of the quantum collapse and creation of a ground state which is missing in the single-particle formulation Sakaguchi and Malomed (2011), while the full many-body analysis demonstrates that the same newly created state exists as a metastable one Astrakharchik and Malomed (2015).

To summarize, we have showed that the dissociation of the 1D matter-wave breather, initiated by the quench from the fundamental soliton, is a purely quantum many-body effect, as all the MF contributions to the dissociation vanish due to the integrability at the MF level. This conclusion opens the way to observe truly quantum many-body effects without leaving the MF range of experimental parameters. We have evaluated the dissociation time corresponding to typical experimental parameters for atomic solitons. The extrapolation of the present results to a larger number of atoms is justified sup by the comparison with recent results produced by truncated Wigner calculations in Ref. Opanchuk and Drummond . Both Opanchuk and Drummond and our work predict a single gradually-expanding cloud, unlike the abrupt formation of two flying apart fragments, predicted in the previous work Streltsov et al. (2008).

We acknowledge financial support provided jointly by the National Science Foundation, through grants PHY-1402249, PHY-1408309, PHY-1607215, and PHY-1607221, and Binational (US-Israel) Science Foundation through grant 2015616, as well as support from the Welch Foundation (grant C-1133), the Army Research Office Multidisciplinary University Research Initiative (grant W911NF-14-1-0003), the Office of Naval Research, and Israel Science Foundation (grant 1287/17). We thank L. D. Carr, P. Drummond, V. Dunjko, and C. Weiss for valuable discussions.

I Calculation of overlaps

The present analysis is based on the exact quantum Bethe-ansatz eigenfunctions Lieb and Liniger (1963); Berezin et al. (1964); Yurovsky et al. (2008) of the Hamiltonian (1) in the main text,

[TABLE]

which are defined in the simplex $-L/2\leq x_{1}\leq x_{2}\leq\cdots\leq x_{N}\leq L/2$ with periodic boundary conditions in the box $\left[-L/2,L/2\right]$. The coefficients Lieb and Liniger (1963); Yurovsky et al. (2008)

[TABLE]

are expressed in terms of rapidities $p_{j}$. Solution (S-1) can describe several bound states (strings) Berezin et al. (1964); McGuire (1964); Yang (1968); Calabrese and Caux (2007) of $N_{i}$ atoms with center-of-mass momenta $P_{i}$. Coefficients $A(\mathcal{P})$ vanish if the permutations $\mathcal{P}$ change ordering within a string. This provides exponential decay of the wavefunction (S-1) when the distance between atoms tends to infinity. For multi-string solutions the normalization factor $\mathcal{N}$ is derived in Ref. Calabrese and Caux (2007).

In the center-of-mass system, the single-string state $\left|\varphi_{N}\right\rangle$ Berezin et al. (1964); McGuire (1964) has rapidities

[TABLE]

and the only non-vanishing coefficient being $A(\mathcal{E})=N!$, where $\mathcal{E}$ is the identity permutation. The normalization factor

[TABLE]

provides $\left\langle\varphi_{N}|\varphi_{N}\right\rangle=1$.

In the case of the double-string state $\left|\varphi_{N_{1}N_{2}v}\right\rangle$, containing $N_{1}$ and $N_{2}=N-N_{1}$ atoms, respectively, in each string, rapidities $p_{j}$ are given by

[TABLE]

Here the string center-of-mass momenta $P_{1}=mvN_{2}/N$ and $P_{2}=-mvN_{1}/N$ are related to the relative string velocity $v$ . The normalization factor,

[TABLE]

is calculated using results from Ref. Calabrese and Caux (2007). It provides $\left\langle\varphi_{N_{1}N_{2}v^{\prime}}|\varphi_{N_{1}N_{2}v}\right\rangle\rightarrow\delta(v-v^{\prime})$ in the limit of $L\rightarrow\infty$. The velocity scale $v_{0}$ is defined by Eq. (3) in the main text.

Rapidities of the initial state $\left|\varphi_{N}^{(0)}\right\rangle$ are given by Eq. (S-3) where $g$ is replaced by $g_{0}=g/4$. The overlap

[TABLE]

calculated using Eq. (S-1), is independent of $L$. Here rapidities $p_{j}$ are given by Eq. (S-5) and normalization factor $\mathcal{N}_{N}^{(0)}$ is given by Eq. (S-4) with $g$ replaced by $g_{0}$. This overlap is used in Eq. (2) in the main text.

Whenever $v=0$ we have $P_{1}=P_{2}$ and Eq. (S-5) gives $p_{j}=p_{j+N/2}$. This means that if $N$ is an even number, i.e. $N_{1}$ and $N_{2}$ have the same parities, the set (S-5) contains equal rapidities. Therefore, coefficients $A(\mathcal{P})$ contain divergent factors at $v=0$. In addition, divergent factors may appear in Eq. (S-6) whenever the sum in the curly brackets vanishes. Divergences are canceled if all terms in Eq. (S-6) are combined with a common denominator. Since the number of terms in the sum over permutations $\mathcal{P}$ between strings may be as large as $\sim 2^{N}$, this transformation was performed using the computer algebra system Maxima max . Nevertheless, the system runs out of available computer resources when $N$ exceeds $20$.

For odd $N$ Eq. (S-6) does not contain divergences. However, due to the loss of significance, the calculations become unreliable for $N>23$.

II Approximate density evolution

The mean-field Gross-Pitaevskii equation has an exact solution corresponding to two scattering solitons Gordon (1983). We are interested in the case when the solitons contain $N_{1}=3N/4$ and $N_{2}=N/4$ atoms and their relative phase and the distance between them are equal to zero at $t=0$, such that the solution coincides with the breather at $t=0$. The mean-field value at the center of mass ($x=0$) is expressed in terms of the scaled time $\tilde{t}=2\pi t/T_{\mathrm{breather}}$ and velocity $\tilde{v}=2v/(Nv_{0})$,

[TABLE]

where $z=-3\tilde{v}\tilde{t}/4$, $\phi=(1+\tilde{v}^{2})\tilde{t}/2$, and $T_{\mathrm{breather}}=32\pi\hbar^{3}/(mg^{2}N^{2})$ is the classical breather period.

Let us approximate the system density by the two-soliton density averaged over their relative velocity $v$,

[TABLE]

The probability distribution can be approximated by a Gaussian,

[TABLE]

where $d_{v}=(\ln 2)/0.44^{2}\approx 3.58$ provides the fit

[TABLE]

[see Eq. (8) in the main text].

The averaged mean-field density $\rho(0,t)$, calculated by numerical integration over $v$ in Eq. (S-7), is compared in Fig. S1 to results produced by truncated Wigner calculations in Ref. Opanchuk and Drummond . The averaged mean-field neglects interference between states with different relative velocities and correlations during evolution. This leads to the underestimated damping of the breather oscillation amplitude. However, the average (over the breather period) values, which decrease due to the breather’s dissociation, are in a good agreement. This justifies the extrapolation of the scaling low (S-8) to large $N$.