Momentum Distribution Dynamics of a Tonks-Girardeau Gas: Bragg Reflections of a Quantum Many-Body Wave Packet

The momentum distribution (MD) dynamics of a Tonks-Girardeau (TG) gas is studied in the context of Bragg reflections of a many-body wave packet. We find strong suppression of a Bragg reflection peak for a large and dense TG wave packet; our observation illustrates the dependence of the MD on the interactions and wave function symmetry. The MD is calculated from the reduced single-particle density matrix (RSPDM). We develop a method for calculating the RSPDM of a TG gas, which is operative for a large number of particles, and does not depend on the external potential and the state of the system. The method is based on a formula expressing the RSPDM via a dynamically evolving single-particle basis.

Figure 1

Dynamics of MDs. (a) $n_F(k,t)$ in the initial stage of the evolution (lower), and after long-time propagation (upper); beating at $k=-\pi /D$ is a signature of Bragg reflections. (b) Initial MDs $n_F(k,0)$ and $n_B(k,0)$ for $N=25$ bosons (solid lines); dot-dashed line depicts $\sum _{m=1}^5|{\tilde{\psi }}_m(k,t){|}^2$ (upper), and $\sum _{i=1}^{12}{\lambda }_i(t)|{\tilde{\varphi }}_i(k,t){|}^2$ (lower) at $t=34.5\mathrm{ms}$ (see text). (c) $n_B(k,t)$ for $N=25$ bosons in the initial evolution stage (lower), and after long-time propagation (upper); the signature of Bragg reflections at $k=-\pi /D$ is absent. (d) Dynamics of $n_B(k,t)$ for $N=3$ bosons; there is beating at $k=-\pi /D$.

Figure 2

Dynamics of bosonic NO occupations ${\lambda }_i$ and entropy $S$ for $N=25$ bosons. (a) ${\lambda }_i(t)$ at times $t=0$, 0.40, 2.15, 33, 34.5 ms; $V_0=11.9\mathrm{peV}$. (b) The entropy $S(t)$ for three different lattice depths $V_0=0$, 4.77, 23.85 peV. (c) $|{\rho }_F(0,x,t)|$ and $|{\rho }_B(0,x,t)|$ at $t=0$ (solid line) and $t=34.5\mathrm{ms}$ (dot-dashed line); $V_0=11.9\mathrm{peV}$. (d) Dynamics of the MD for an ideal Bose gas. See text for details.