Single- to many-body crossover of a quantum carpet

A strongly interacting many-body system of bosons exhibiting the quantum carpet pattern is investigated exactly by using Gaudin solutions. We show that this highly coherent design usually present in noninteracting, single-body scenarios gets destroyed by weak to moderate interatomic interactions in an ultracold bosonic gas trapped in a box potential. However, it becomes revived in a very strongly interacting regime, when the system undergoes fermionization. We track the whole single- to many-body crossover, providing an analysis of de- and rephasing present in the system.

Figure 1

Top row: Quantum carpets of $N=3$ bosons which are (a) ideal, interacting repulsively with (b) $\gamma =0.86$ or (c) $\gamma =20.0$, or (d) fermionized. The time coherence is preserved in the first and last cases, while a finite interaction smears out the pattern, signaling loss of coherence in the system. Bottom row: The squares of the autocorrelation functions for each of these cases. Perfect revivals are present at the multiples of the revival time $T_{\text{rev}}$ for ideal and fermionized bosons. Imperfect revivals are visible for intermediate interaction strengths, being shifted away from $T_{\text{rev}}$ [see (c)].

Figure 2

Momentum quantum carpets of $N=2$ bosons which are (a) ideal, (b) interacting repulsively with $\gamma =5.8$, or (c) fermionized. The last two carpets present an evolution of the momentum distribution for (d) $N=2$ and (e) $N=5$ ideal fermions. Note that there is a visible difference between fermionized bosons and ideal fermions, contrary to the quantum carpet in the position space. Once again, the loss of temporal coherence due to finite interactions is apparent in (b).

Figure 3

Modulus of reduced single-particle density matrices for $N=2$ bosons that are (a) ideal, (b) interacting repulsively with $\gamma =22.5$, or (c) fermionized at time $t=0.0914T_{\text{rev}}$ and for (d) $N=5$ fermionized bosons or (e) ideal fermions at time $t=0.0725T_{\text{rev}}$. The spatial coherence is present all across the system in the ideal bosonic and ideal fermionic cases; however, the structure of coherence is visibly different—in the latter case rectangular-shaped, narrow structures are present, signifying correlations between solitonlike canals and ridges from the fermionic quantum carpet. Fermionized bosons do not recreate this type of coherence, with off-diagonal parts decaying as one goes away from the diagonal.

Figure 4

(a) Maximum of a square of an autocorrelation function, $A_{\text{max}}$, close to the revival time $T_{\text{rev}}$ for $N=2$ and $N=3$ bosons for different interaction strengths. The decrease in the value signifies the imperfect revival. Left (two atoms) and right (three atoms) insets show a density plot of $A\left(t\right)$ close to $T_{\text{rev}}$. One can see that the revival time does not change for a weak interaction, while it increases with the diminishing of interaction strength close to fermionization. Additionally, for intermediate interactions, a distinctive peak of the square of the autocorrelation function is visible. Black curves are computed from the asymptotic formula (9). (b) Entanglement entropy averaged over the interval $t\in [0,T_{\text{rev}}], {\langle \mathcal{E}\rangle }_{T_{\text{rev}}}$ for $N=2$ bosons. The right inset provides the short-time behavior of the entanglement entropy. Interactions induce the growth of entropy in time. For strong enough interactions entropy saturates at some value that does not change further. In the fermionized regime, the average value of the entropy does not change in time with minimal values reached for multiples of $T_{\text{rev}}/2$. The left inset shows the evolution of entropy divided by its initial value for different interaction strengths. We observe the rapid growth of that quantity for intermediate interactions for which time coherence is lost over time. One can see that interaction introduces temporal incoherence in the system, which is later suppressed while entering the coherent, fermionized regime.

Figure 5

Variance of $\text{dist}\left(\mathrm{\Delta }E\right)$ (see the text) for $N=3$ bosons as a function of the interaction, with histograms illustrating different regimes. For weak and strong interactions, the histograms are very narrow, and the corresponding variance is almost zero, as the energies of the system are close to being commensurate. The nonzero values of variance correspond to energy differences deviating from the form $\text{(integer)}\times 2\pi \frac{\hslash }{T_{\text{rev}}}$. Such a property of the spectrum affects the temporal coherence of the system by destroying perfect revivals and shifting the revival time. Around $\gamma \sim 1.5$ we observe a small peak, followed by a plateau at the level given by the variance of the uniform distribution.