Single-particle density matrix and the momentum distribution of dark “solitons” in a Tonks-Girardeau gas

We study the reduced single-particle density matrix (RSPDM), the momentum distribution, and natural orbitals and their occupancies of dark “soliton” (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points ($x$ and $-x$) with respect to the dark notch at $x=0$; the correlations oscillate rather than decay as the points $x$ and $-x$ are being separated.

Figure 1

(Color online) The RSPDM of dark soliton states, ${\rho }_{\mathit{DS}}(x,y)$, for $N=5$ (a), 11 (b), 17 (c), and 25 (d).

Figure 2

(Color online) The momentum distributions corresponding to DS states (a) and to degenerate eigenstates with randomly chosen phases (b); figures are shown for $N=11$ (× symbols, blue dotted line), 17 (diamonds, red dashed line), and 25 (circles, solid black line).

Figure 3

(Color online) The cross-diagonal ${\rho }_{\mathit{DS}}(x,-x)$ of the dark soliton RSPDM (solid black line), displaying long-range oscillatory correlations between mirror points $x$ and $-x$, and the cosine function $\cos \left(2k_{\mathit{\text{peak}}}x\right)$ (red dotted line).

Figure 4

(Color online) The RSPDM of typical eigenstates ${\psi }_{\mathit{DE}}$ [Eq. (23)] for $N=7$ (a), 11 (b), 17 (c), and 25 (d).

Figure 5

(Color online) The occupancies of the NOs for the state ${\psi }_{\mathit{DS}}$ (black diamonds) and a typical state ${\psi }_{\mathit{DE}}$ (red squares), for $N=25$ particles. The sharp drop in the occupancies of ${\psi }_{\mathit{DS}}$ occurs between ${\lambda }_N$ and ${\lambda }_{N+1}$.

Figure 6

(Color online) The NOs of a DS state in $x$ space (left column) and their power spectra in $k$ space (right column) for $N=25$. (a) The first ${\Phi }_1\left(x\right)$ (black solid line) and third ${\Phi }_3\left(x\right)$ (red dotted line) NOs. (b) The Fourier power spectrum of the first ${\mid {\tilde{\Phi }}_1\left(k\right)\mid }^2$ (black circles, solid line) and third ${\mid {\tilde{\Phi }}_3\left(k\right)\mid }^2$ (red diamonds, dotted line) NOs. (c) The 24th (red dotted line) and 25th (black solid line) NOs in $x$ space. (d) The momentum distribution (red squares, dotted line) in comparison to the contribution from the 24th and 25th NOs: ${\sum }_{i=24}^{25}{\lambda }_i{\mid {\tilde{\Phi }}_i\left(k\right)\mid }^2$ (black circles, solid line).

Figure 7

Momentum distributions for ten different states ${\psi }_{\mathit{DE}}$ [Eq. (23)] chosen at random, by randomly choosing ten sets $\{{\theta }_m\mid m=1,\dots ,N\}$ of phases (see text for details).

Figure 8

(Color online) The RSPDM (a), momentum distribution (b), and occupancies of the NOs (c) for a DS state $(N=10)$ in a parity-invariant well-shaped potential. (d) The momentum distribution (black dotted line) in comparison to the contribution from the ninth and tenth NOs: ${\sum }_{i=9}^{10}{\lambda }_i{\mid {\tilde{\Phi }}_i\left(k\right)\mid }^2$ (red solid line).