Mesoscopic superpositions of Tonks-Girardeau states and the Bose-Fermi mapping

We study a one-dimensional gas of repulsively interacting ultracold bosons trapped in a double-well potential as the atom-atom interactions are tuned from zero to infinity. We concentrate on the properties of the excited states which evolve from the so-called NOON states to the NOON Tonks-Girardeau states. The relation between the latter and the Bose-Fermi mapping limit is explored. We state under which conditions NOON Tonks-Girardeau states, which are not predicted by the Bose-Fermi mapping, will appear in the spectrum.

Figure 1

(a) The first ten eigenvalues of the single-particle Hamiltonian as a function of $V_0$. The dashed line represents the condition $E=V_0$. (b)–(d) One-body density matrix for NOON Tonks-Girardeau states of $N=2,3$, and 4 obtained with $g=20\mathrm{and}{V}_0=6,10$, and 12, respectively.

Figure 2

Energy spectra for (a) $N=2$ with $V_0=5$ and (b) $N=3$ with $V_0=10$. The blue dashed lines represent the energies expected in the fermionized limit, $g\to \infty$ (we explicitly label a few of them). We extrapolate these energies from the ones calculated with an increasing number of modes, with a maximum of 30 (24) modes for $N=2$ (3).

Figure 3

One-body density matrices of the first and third excited states for two bosons and $V_0=5$. The top (bottom) row is computed with $g=0.5$ $\left(g=20\right)$. The NOON state in (a) evolves into the NOON TG in (c). The noninteracting state in (b) remains mostly unchanged in (d).

Figure 4

Comparison of two of the four excited numerical wave functions for $g=20$ (top row) and the analytic ones ${\mathrm{\Psi }}_1^{\mathrm{b}}$ [(e) and (g)] and ${\mathrm{\Psi }}_3^{\mathrm{b}}$ [(f) and (h)] obtained through the BF mapping (bottom row). (a), (b), (e), and (f) are obtained with $V_0=4$, and (c), (d), (g), and (h) are obtained with $V_0=5$. $N=2$.

Figure 5

Detailed analysis of the $g\to \infty$ limit for $V_0=5.5$ and $N=2$ (top panel). Two of the first excited states computed numerically (solid lines) are compared to the results of the perturbation theory when the width of the perturbation is $\sigma \simeq 0.07$ (dashed lines). The bottom panels depict the composition of the dressed states, ${\overline{\mathrm{\Psi }}}_1^b$ and ${\overline{\mathrm{\Psi }}}_4^b$: $\langle {\overline{\mathrm{\Psi }}}_k^b|{\mathrm{\Psi }}_1^b\rangle$ (solid red line) and $\langle {\overline{\mathrm{\Psi }}}_k^b|{\mathrm{\Psi }}_4^b\rangle$ (dashed blue line). A similar qualitative picture is obtained for states ${\overline{\mathrm{\Psi }}}_2^b$ and ${\overline{\mathrm{\Psi }}}_3^b$.

Figure 6

(a) Critical interaction strength $g_c$ for different values of $\sigma$ and different numbers of modes $M$. (b) Critical interaction strength for different values of $\sigma$. The dotted line is a fit of the form $g_c=b_0+b_1/\sigma$ to the four lower values of $\sigma =0.05,0.07,0.1$, and $0.15$.