Ground-state properties of a Tonks-Girardeau gas in a split trap

We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\delta$-function barrier at the trap center. This is done by calculating the reduced single-particle density matrix, the pair-distribution function, and the momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground-state occupation in a diagonal basis diverges from the $\sqrt{N}$ behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.

Figure 1

(Color online) Reduced single-particle density matrices $\rho \left(x,x^{\prime }\right)$ for a Tonks gas of twenty particles for splitting strengths $\kappa =0$ (left) and $\kappa =20$ (right).

Figure 2

Single-particle density for a Tonks gas of twenty particles for increasing barrier height.

Figure 3

The first three natural orbitals for a twenty particle Tonks gas in a $\delta$-split trap with splitting strength $\kappa =20$.

Figure 4

(Color online) Pair distribution functions $D\left(x_1,x_2\right)$ for a Tonks gas of $N=5,10$, and 30 particles in $\delta$-split trap for splitting strengths $\kappa =0,1$, and 10. In each plot the horizontal and vertical axes run from $-10$ to $+10$ in scaled units.

Figure 5

Plot of the fractional ground-state occupation $f=\frac{{\lambda }_0}{N}$ versus particle number for a TG gas in a $\delta$-split trap with increasing splitting strength.

Figure 6

(Color online) Central peaks of the momentum distributions $n\left(k\right)$ for TG gases consisting of $N=5$ (left) and $N=6$ (right) particles, for values of splitting strength $\kappa =0,10$, and 20.

Figure 7

(Color online) Central peak of the momentum distribution $n\left(k\right)$ for TG gases consisting of $N=19$ (left) and $N=20$ (right) particles, for values of splitting strength $\kappa =0,10$, and 20.

Figure 8

Magnitude of the first six eigenvalues ${\lambda }_j$ of $\rho \left(x,x^{\prime }\right)$ for $N=19$ and 20 particles as a function of the splitting strength $\kappa$. The lines of greatest magnitude represent ${\lambda }_0$ and decrease in the order ${\lambda }_0,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$.

Figure 9

(Color online) Free temporal evolution of the single-particle density $\rho \left(x,t\right)$ for a $N=5$ and 6 particle TG gas initially confined in a $\delta$-split trap with $\kappa =100$.

Figure 10

(Color online) Free temporal evolution of the single-particle density $\rho \left(x,t\right)$ for a $N=19$ and 20 particle TG gas initially confined in a $\delta$-split trap with $\kappa =100$.