 0  0.2  0.4  0.6  0.8  1  1.2  1.4  100  200  300  400  500  600 Trapping energy, U relax. /U therm. Energy, E//h perp * l n l n l n l n 3 l n  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0  50 100 150 200 250 300 350 Occupation probability, P(l) Longitudinal mode, l beamsplitter initial relaxed eigenstate microcanon. 10 (b) l n  0  0.2  0.4  0.6  0.8  1  1.2  0  20  40  60  80  100 Occupation probability, P(n) Transverse mode, n initial relaxed eigenstate microcanon. 100 (a) FIG. 1. The time average of the transverse trapping energy U$_{relax.}$ ≡ lim t$_{max}$ →∞ (1 /t max ) ∫ t$_{max}$ 0 〈 ψ ( t ) | ˆ U | ψ ( t ) 〉 after relax- ation from an initial state as a function of the energy of the initial state E [see Hamiltonian (1)]. Four families of the ini- tial states are considered. Long-dashed (green) line: state (5) with n 0 = 0, δ = 0 . 99, and the energy being controlled via scanning l 0 ; short-dashed (blue) line: the same as the previ- ous one, except for δ = . 1; dotted (purple) line: the state (5) with δ = 0 . 1 and l 0 = 0 and the energy control via n 0 ; solid (red) line: state (6) with κ 2 = 2 κ 1 , δ = 0 . 99, l 0 = 0 and the energy control via κ 1 . Legends schematically show the distri- bution of the initial states over the quantum numbers; arrows indicate the direction in which the parameters controlling the energy are scanned. The asterisk (*) corresponds to the set of parameters used to produce the Fig. 2. Grey dots show the quantum expectation value of the trapping energy U for every hundreds eigenstate of the full Hamiltonian (1,2). All the values of the trapping energy are shown with respect to its microcanonical expectation value U$_{therm.}$ ≈ E/ 3. For all the points, ( L/a ⊥ ) 2 = 2 φ gr π 7 ≈ 9774 and a s = 10 $^{6}$a ⊥ . FIG. 2. Distribution of the mean occupation numbers for the transverse (a) and longitudinal (b) modes of the unper- turbed Hamiltonian. The long-dashed (green) and solid (red) lines show the initial and time-averaged distributions for an initial state (5) with n 0 = 0, δ = 0 . 99, and l 0 = 292 . 8. En- ergy of the initial state is E ≡ (2 /λ )¯ hω ⊥ ϵ α = 173 . 2¯ hω ⊥ . The short-dashed (blue) line corresponds to the 17096’s eigenstate (closest in energy to E from the above, with a relative dis- crepancy of 10 − $^{5}$) of the interacting Hamiltonian. The dotted (purple) line shows the microcanonical prediction. The rest of the parameters is the same as at the Fig. 1. 〈 z | ψ ( t =0) 〉 ∝ cos πζ δ θ ( δ 2 − | ζ | ) cos (2 πl$_{0}$ζ ) | n$_{0}$ 〉 . (5) thermal prediction. Consider the following initial state: 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]. 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. 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