Collision of one-dimensional fermion clusters

We study cluster-cluster collisions in one-dimensional Fermi systems with particular emphasis on the nontrivial quantum effects of the collision dynamics. We adopt the Fermi-Hubbard model and the time-dependent density-matrix renormalization-group method to simulate collision dynamics between two fermion clusters of different spin states with contact interaction. It is elucidated that the quantum effects become extremely strong with the interaction strength, leading to the transmittance being much more enhanced than expected from the semiclassical approximation. We propose a concise model as an application of the Bethe ansatz, which unveils the origin of the quantum effects and also explains the overall properties of the simulation results clearly. This model provides an intuitive perspective of the collision dynamics with contact interaction. Some potential applications, such as repeated collisions, are addressed.

Figure 1

Initial particle density and trap potential at $n=6$ and initial spin-down density at $n=1$, $n=2$, and $n=3$.

Figure 2

(a) Reflectance obtained by DMRG simulation $R_n\left(u\right)$ (dots) and that by ICM $R_n^{\mathrm{ICM}}\left(u\right)$ (lines) (log scale on $x$ and $y$ axes). (b) Reflectance ratio obtained by DMRG simulation $R_n\left(u\right)/R_1\left(u\right)$ (dots) and that by ICM $R_n^{\mathrm{ICM}}\left(u\right)/R_1^{\mathrm{ICM}}\left(u\right)$ (lines) (log scale on $x$ axis).

Figure 3

(a) Transmittance obtained by DMRG simulation $T_n\left(u\right)$ (dots) and that by ICM $T_n^{\mathrm{ICM}}\left(u\right)$ (lines) (log scale on $x$ and $y$ axes). (b) Transmittance ratio obtained by DMRG simulation $T_n\left(u\right)/T_1\left(u\right)$ (dots) and that by ICM $T_n^{\mathrm{ICM}}\left(u\right)/T_1^{\mathrm{ICM}}\left(u\right)$ (lines) (log scale on $x$ axis).

Figure 4

Collision order in ICM at $n=3$. $S_{-ij}$ is the one-to-one collision between ${\varphi }_{-i}$ and ${\varphi }_j$, and ${\psi }_k$ is the wave function after the $k$th set of collisions. Simultaneous collisions are commutative.

Figure 5

Initial particle density and trap potential at $n=2$ and $d=4\eta$.