Dynamical Quasicondensation of Hard-Core Bosons at Finite Momenta

Long-range order in quantum many-body systems is usually associated with equilibrium situations. Here, we experimentally investigate the quasicondensation of strongly interacting bosons at finite momenta in a far-from-equilibrium case. We prepare an inhomogeneous initial state consisting of one-dimensional Mott insulators in the center of otherwise empty one-dimensional chains in an optical lattice with a lattice constant $d$. After suddenly quenching the trapping potential to zero, we observe the onset of coherence in spontaneously forming quasicondensates in the lattice. Remarkably, the emerging phase order differs from the ground-state order and is characterized by peaks at finite momenta $\pm (\pi /2)(\hslash /d)$ in the momentum distribution function.

Figure 1

Quasicondensation of bosons. (a) Density of states $\rho (\epsilon )$ of a homogeneous 1D lattice. (b) Dispersion $ϵ(q)$ (solid line) and group velocity $v_g(q)$ (dotted line) versus quasimomentum $q$. (c) Sketch of quasimomentum distribution $n(q)$: In equilibrium, 1D bosons quasicondense at the minimum of the band at $q=0$, while in a sudden expansion the quasicondensation of hard-core bosons occurs at $\hslash q=\pm (\pi /2)(\hslash /d)$. This quasimomentum lies in the middle of the spectrum and is consistent with the vanishing energy per particle of this closed many-body system.

Figure 2

Finite-momentum quasicondensation under ideal conditions [28]. We consider $N=50$ initially localized HCBs. (a) Density $n_j$ as a function of time. (b) Density as a function of the rescaled coordinate $\tilde{x}=(j-25.5)/(2t_E/\tau )$ in the region bounded by the dashed lines in (a). For $\tilde{x}\le 1$, the data collapse to the scaling solution [34] $n(\tilde{x})=\arccos (\tilde{x})/\pi$. (c) Quasimomentum distribution $n(q)$ as a function of time. (d) One-particle correlations at $t_E=0.24N\tau$. Main panel: $|⟨{\widehat{a}}_j^{†}{\widehat{a}}_{j+r}⟩|$ at $j=26$ (circles) and $\mathcal{A}(r)=\alpha /\sqrt{r}$ (line) with $\alpha =0.29$ [35]. Inset: Phase pattern $\mathrm{\Phi }(r)$.

Figure 3

Dynamics during sudden expansion and time-of-flight sequence. (a) Sketch of the experimental sequence: We start with a trapped gas that is released from the initial confinement at $t=0$ (top) and then expands for a time $t_E$ in the optical lattice (middle). At $t_E$, all potentials are removed and the distribution is measured after a finite time-of-flight time $t_{\mathrm{TOF}}$ (bottom). (b)–(e): In situ density distributions during the sudden expansion integrated along the $y$ and $z$ axes. The lattice constant corresponds to $d=\lambda /2\approx 368\mathrm{nm}$. (f)–(i): TOF density distributions taken at $t_{\mathrm{TOF}}=6\mathrm{ms}$ for the same $t_E$ as in (b)–(e). All measured density distributions are averaged over 9 to 11 experimental realizations.