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thermal prediction. 

Consider the following initial state: 

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Longitudinally, the state is represented by the ground state of a length Lδ hard-wall box split initially by an ideal beamsplitter with momenta ± 2 πl$_{0}$/L ; the box is centered at the maximal interatomic distance. The state is distributed among approximately π/δ axial modes lo- calized about l = ± l$_{0}$ . The initial transverse state is limited to a single mode n$_{0}$ . Note that for n$_{0}$ = 0 and δ ≈ 1 the state is conceptually similar to the initial state used in the equilibration experiments [1]. 

After relaxation, both transverse (Fig. 2a) and longi- tudinal (Fig. 2b) distributions remain very far from the equilibrium predictions. Fig. 1 shows the equilibrium value of the transverse trapping energy U (see (1)) for three sequences of the initial states of the type (5). The first two, with low transverse energy, show an approx- imately -40% deviation of the relaxed value of U from 

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the thermal prediction, thus being seemingly correlated with its initial, negative value. Note that the correla- tion between the final and initial values of observables in quantum systems is addressed in Ref. [28]. Also, note that the initial states of the second sequence have a much greater spread over the longitudinal modes than the first one; consistently, the energy-to-energy variation of the relaxed values of U is less than for the first sequence. The third sequence, with low longitudial energy, shows devi- ations from the thermal prediction that range between +50% and +5% but never reach zero. 

