Matter-wave scattering from interacting bosons in an optical lattice

We study the scattering of matter waves from interacting bosons in a one-dimensional optical lattice, described by the Bose-Hubbard Hamiltonian. We derive analytically a formula for the inelastic cross section as a function of the atomic interaction in the lattice, employing Bogoliubov's formalism for small condensate depletion. A linear decay of the inelastic cross section for weak interaction, independent of the number of particles, condensate depletion, and system size, is found.

Figure 1

Scattering setup: A particle of mass $m$ initially in a plane-wave state with momentum ${\mathit{k}}_0$ is scattered into the angle $\theta$ from a target of atoms (all of which have mass $M$) submerged in a one-dimensional optical lattice with depth $V_0$ and lattice constant $d=\pi /k_L$, where $k_L$ is the laser wave number. The asymptotic final state of the probe has momentum $\mathit{k}$.

Figure 2

Full scattering cross section normalized by $N^2$ in the SF and MI limits, from the expressions given in Table 1 using the harmonic approximation for the Wannier functions [see Eq. (45)], for $N=9$ particles in a lattice of length $L=9$. The scattering signal is symmetric around $\theta =0$. The incoming energy is $E_0=2E_r$, the hopping energy is set to $J=0.0065E_r$, and the masses of the probe and target atoms are taken to be equal. The elastic part of the cross section is the same in both regimes. The shaded region highlights the inelastic scattering component. The inset shows the inelastic cross section normalized by $N$, obtained from Eq. (4) and the numerically calculated spectrum of the system, for different values of the interaction $U$.

Figure 3

Relative condensate depletion $\delta n/n$ as a function of the interaction parameter $\mathcal{U}=Un/J$, for $N=20$ particles on $L=5$ sites. The inset shows the behavior of $\delta n/n$ for small $\mathcal{U}$ (black solid line) and the quadratic approximation (red dashed line), Eq. (37).

Figure 4

Inelastic cross section as a function of the scattering angle $\theta$ for $N=25$ particles in $L=5$ sites, normalized by $N$, for different interaction strength $\mathcal{U}$. The incoming energy is $E_0=2E_r$, the hopping energy is set to $J=0.0065E_r$, and we use $m=M$. Symbols correspond to the exact cross section obtained from the diagonalization of $H_{\mathrm{BH}}$. Red lines show the Bogoliubov approximated cross section [Eq. (44)]. The quality of the approximation is independent of the angle.

Figure 5

Dependence of the inelastic Bogoliubov cross section on the incoming energy $E_0$ and the scattering angle $\theta$ for $N=100$ particles on $L=100$ sites at fixed interaction strength $U=0.02J$, for which $\delta n/n=0.012$. We set $J=0.0065E_r$ and $m=M$.

Figure 6

Comparison of the large-$L$ formula of the Bogoliubov cross section [Eq. (47), black lines] vs expression (44) (orange thick lines) for (a) $L=10$, $n=10$, $E_0=2E_r$, $U=2J$ ($\delta n/n=0.080$); (b) $L=100$, $n=5$, $E_0=3E_r$, $U=0.5J$ ($\delta n/n=0.098$); and (c) $L=1000$, $n=1$, $E_0=5E_r$, $U=0.01J$ ($\delta n/n=0.027$). In all cases $J=0.0065E_r$ and $m=M$.

Figure 7

Main panel: Inelastic cross section into the angle $\theta =\pi /4$ as a function of $\mathcal{U}=Un/J$ for $N=10$ (red diamonds) and $N=25$ (black circles) bosons in a lattice of length $L=5$. The incoming energy is $E_0=2E_r$, the hopping energy is set to $J=0.0065E_r$, and we use $m=M$. Solid lines correspond to the analytical formula (44), while symbols are exact results. The inset shows the behavior for small values of $\mathcal{U}$. For $\mathcal{U}\ll 1$, the normalized inelastic cross section becomes independent of the density $n$, and can be approximated by a linear function (blue dashed line) given by Eq. (52). Lower panel: Relative condensate depletion for $N=10$ (red dashed line) and $N=25$ (black solid line) particles.

Figure 8

Average relative deviation ${\Delta }_{\mathrm{CS}}$ of the Bogoliubov inelastic cross section with respect to the exact result, as a function of the interaction $U/J$ and the density $n$ for a system of $L=5$ sites (note the inverted $x$ axis). Relevant parameters are the same as in Fig. 7. Contour lines corresponding to $3\%$ (gray dashed line), $10\%$ (white dashed line), and $30\%$ (black solid line) deviation are marked. The inset shows the relative condensate depletion $\delta n/n$.

Figure 9

Slope $\Lambda (L,E_0,\theta )$ of the decay of the inelastic Bogoliubov cross section at incoming energy $E_0=5E_r$, $J=0.0065E_r$, and $m=M$, for $L=10$ (blue line), $L=100$ (orange line), and the large-$L$ approximation (dashed line) from Eq. (53). White circles highlight the angles at which the large-$L$ approximation is exact for $L=10$, given in Eq. (A13). The lower panel shows the inelastic cross section for $L=100$ and $n=5$ at $U=0$ and $U=0.01J$ ($\delta n/n=5\times {10}^{-3}$).

Figure 10

Decay of the inelastic cross section vs $\mathcal{U}$, for $E_0=5E_r$, different scattering angles, system sizes, and densities, as indicated in the plot. Solid lines are obtained from the Bogoliubov formula (44) and black dashed lines depict the linear approximation given in Eq. (53). At $\mathcal{U}=1$, we have $\delta n/n=0.011$ for $L=1000$ and $n=50$, and $\delta n/n=0.054$ for $L=100$ and $n=5$. The inset shows exact numerical results (symbols) obtained for $E_0=3E_r$, and blue dashed lines correspond to the linear decay given by Eq. (52). In all cases $J=0.0065E_r$ and $m=M$.

Figure 11

Elastic cross section [Eq. (A1), red line] and inelastic cross section in the SF limit [Eq. (A4), black line] for a lattice of size $L=50$ with $E_0=5E_r$, $V_0=15E_r$, $J=0.0065E_r$, and $m=M$. The thick orange line shows the form factor $|W\left({\kappa }_{\mathrm{el}}\right){|}^2$, which determines both cross sections for large $L$ according to Eqs. (A3) and (A9). Note the different normalization factors: In a system with $N$ bosons, the elastic cross section in units of $a_s^2$ is $N^2{\Gamma }_{\mathrm{el}}$ and the inelastic $N{\Gamma }_{\mathrm{inel}}^{\mathrm{sf}}$. Vertical dashed lines mark the intervals given in Eq. (A12).