Symmetry Breaking and Enhanced Condensate Fraction in a Matter-Wave Bright Soliton

An exact diagonalization study reveals that a matter-wave bright soliton and the Goldstone mode are simultaneously created in a quasi-one-dimensional attractive Bose-Einstein condensate by superpositions of quasidegenerate low-lying many-body states. Upon formation of the soliton the maximum eigenvalue of the single-particle density matrix increases dramatically, indicating that a fragmented condensate converts into a single condensate as a consequence of the breaking of translation symmetry.

Figure 1

(a) Excitation spectrum obtained by exact diagonalization of Hamiltonian (1) for $N=200$. The inset shows the corresponding result obtained with truncation $l_{\mathrm{c}}=2$ near the critical point. (b) Bogoliubov spectrum corresponding to (a), where branch $A^{\prime }$ represents the Goldstone mode, $B^{\prime }$ the breathing mode of a bright soliton, and $C^{\prime }$ the second harmonic of $B^{\prime }$. (c) Energy gap $\Delta E$ between the ground and the first excited states in the many-body spectrum versus the total number of atoms $N$ with $gN=1.4$ held fixed. Triangles and circles denote results obtained with $l_{\mathrm{c}}=1$ and $l_{\mathrm{c}}=2$, respectively.

Figure 2

Low-lying spectrum (thin lines, left scale) and expectation value of the total angular-momentum (thick line, right scale) of the ground state versus the angular frequency of the rotating drive for $g=2\times {10}^{-3}$ and $N=100$. (b) is an enlargement of (a).

Figure 3

The largest eigenvalue of the reduced single-particle density matrix obtained by the exact diagonalization (${\lambda }_{\mathrm{M}}^{(\epsilon )}$) and Bogoliubov theory (${\lambda }_{\mathrm{B}}^{(\epsilon )}$) versus $\epsilon N^2$ for $gN=1.4$. Inset: Distribution of $|{\beta }_{\mathcal{L}}{|}^2={|}_A⟨\mathcal{L}|\Psi (\epsilon )⟩{|}^2$ with $\epsilon =1\times {10}^{-4}$. The solid curves depict Eq. (4) with Eq. (5).

Figure 4

Ensemble-averaged largest eigenvalue ${\overline{\lambda }}_{\mathrm{M}}^{(j)}$ of the reduced single-particle density matrix versus the number of quantum measurements $j$ for $g{N}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=1.6$. The solid curve denotes the analytical result (6). Inset: Ensemble-averaged value of $|{\beta }_{\mathcal{L}}{|}^2={|}_A⟨\mathcal{L}|{\Psi }^{(j)}⟩{|}^2$ after $j=5,20$, and $50$ time measurements. The solid curves depict Eq. (4) with $d^2=4j(g{N}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}-1)/(7g{N}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}})$.