Quenches from bosonic Gaussian initial states to the Tonks-Girardeau limit: Stationary states and effects of a confining potential

We consider the general problem of quenching an interacting Bose gas from the noninteracting regime to the strongly repulsive limit described by the Tonks-Girardeau gas with the initial state being a Gaussian ensemble in terms of the bosons. A generic multipoint correlation function in the steady state can be described fully in terms of a Fredholm-like determinant suitable both for a numerical and for an analytical study in certain limiting cases. Finally, we extend the study to the presence of a smooth confining potential showing that, in the thermodynamic limit, the time evolution of the two-point function can be mapped to a classical problem.

Figure 1

Feynman diagram representation of Eqs. (10) and (11): The black dots represent the density operators $n\left(z\right)$. The Wick theorem contracts the fields pairwise: Since the density contains two fields $n\left(z\right)={\psi }^{†}\left(z\right)\psi \left(z\right)$ there are two departing legs from each black dot. In Eq. (10) external legs are present in association with the fields ${\psi }^{†}\left(y\right)$ and $\psi \left(x\right)$ (left), whereas in the diagrams of Eq. (11) external legs are absent (right).

Figure 2

(Left) Stationary fermionic two-point correlation function after an interaction quench ($c=0\to c=\infty$) of a Bose gas initially prepared in a thermal state with inverse temperature $\beta$. Symbols are the numerical evaluation of Eq. (13) with the initial bosonic correlator given by (15). In the limit of zero temperature, the correlation function approaches the analytic result (17) (the full red line). However, for any finite temperature, the long-distance exponential decay is captured by the asymptotic expansion (21) (the dashed lines). (Center) Stationary momentum density distribution of the fermions obtained by Fourier transform $\langle {\mathrm{\Phi }}^{†}\left(z\right)\mathrm{\Phi }\left(0\right)\rangle$. (Right) Stationary bosonic two-point correlation function obtained by numerical evaluation of Eq. (14) with the initial fermionic correlator given by $\langle {\mathrm{\Phi }}^{†}\left(z\right)\mathrm{\Phi }\left(0\right)\rangle$. Notice the long-distance exponential decay. Nevertheless, such a decay is not exponential for all distances. Indeed, by comparing the correlator with the exact exponential decay which matches the short-distance expansion, namely, $n{e}^{-4n^2\left|x\right|}$ (the dot-dashed black line), one reveals an appreciable discrepancy (see the inset for a zoom on the short distances).

Figure 3

Color density plot of the Wigner distribution $F_{\tau }(p,q)$ vs $(q,p)\in [-4,4]\times [-8,8]$ for different times $\tau$ and two representative traps (here $m=1$ and $n=1$). The system initially is prepared at zero temperature with $F_0(p,q)$ given by Eq. (32). In the harmonic case each phase-space trajectory is characterized by the same period thus preserving the shape of the initial Wigner distribution whose support simply rotates in time. In the anharmonic case, the typical dynamical period $T\left(p_0\right)\simeq {\int }_0^{q^{*}}dq/\sqrt{p_0^2-2mV\left(q\right)}$ [with $q^{*}$ being the real positive root of $V\left(q^{*}\right)=p_0^2/2m$] depends on the initial energy, and the $\delta$ support of the Wigner distribution folds over time. For long times, it tends to cover the entire phase plane.

Figure 4

Zero-temperature two-point correlation function for different times after an interaction quench in the anharmonic trap. In the long-time limit the correlator approaches the steady-state prediction of Eq. (30) (the black dashed lines).