$T$-matrix approach to the phonon-mediated Casimir interaction

We develop a theory of the phonon-mediated Casimir interaction between two pointlike impurities which is based on the single-impurity scattering $T$-matrix approach. We show that the Casimir interaction at $T=0$ falls off as a power law with the distance between the impurities. We find that the power in the weak and unitary phonon-impurity scattering limits differs, and we relate the power law to the low-energy properties of the single-impurity scattering $T$ matrix. In addition, we consider the Casimir interaction at finite temperature and show that at finite temperatures the Casimir interaction becomes exponential at large distances.

Figure 1

Diagrammatic representation of the derivative of the thermodynamic potential with respect to the distance of the impurities.

Figure 2

Definition of the single-particle $T$ matrix $T_1\left({\omega }_n\right)$.

Figure 3

Diagrammatic representation of the two-impurity $T$ matrix. The empty circles here correspond to $T_1\left({\omega }_n\right)$, light blue and dark blue circles mean $T_2\left(\frac{\mathbf{r}}{2},\frac{\mathbf{r}}{2},{\omega }_n\right)$ and $T_2\left(-\frac{\mathbf{r}}{2},\frac{\mathbf{r}}{2},{\omega }_n\right)$, respectively, and the wavy lines are the Green's functions $G_r^{\left(0\right)}\left({\omega }_n\right)$.

Figure 4

$U_{\mathrm{Cas}}\left(r\right)$ in the second order of the perturbation theory.

Figure 5

Casimir interaction of dynamic impurities at finite temperature and at $T=0$. $r_T$ is the effective length of the potential. In units of energy, (a) 1D system with parameter $g=0.95$. Red line: $T=0.002{\omega }_0$. Blue line: $T=0$. (b) Two-dimensional system with $g=0.99$. Red line: $T=0.003$. Blue line: $T=0$. ${\lambda }_T\simeq 53$ is the de Broglie wavelength, $r_T\simeq 110$. (c) Three-dimensional system with $g=0.999$. Red line: $T=0.004$. Blue line: $T=0$.

Figure 6

Casimir interaction of dynamic impurities in the unitary limit $\left(g=1\right)$. (a) One-dimensional system. Red line: $T=0.002$. Blue line: $T=0$. (b) Two-dimensional system. Red line: $T=0.003$. Blue line: $T=0$. (c) Three-dimensional system. Red line: $T=0.004$. Blue line: $T=0$.

Figure 7

Casimir interaction for atoms trapped by the external potential at a finite temperature and at $T=0$. $r_T$ is the effective length of the interaction, $g=2,{\omega }_0=1$. Blue line: $T=0$, red line: $T=0.0002$. (a) One-dimensional system. (b) Two-dimensional system. (c) Three-dimensional system.

Figure 8

The ratio of the 1D Casimir interaction in the continuum limit $U_{\mathrm{cont}}$ and the lattice model $U_{\mathrm{lattice}}$. Here $T=0$. (a) Dynamic impurities. Red line, $g=0.01$; blue line, $g=0.1$; yellow line, $g=0.25$; green line, $g=1$. (b) Impurities localized in the external potential. Red line, $g=0.01$; blue line, $g=1$; yellow line, $g=10$; green line, $g\to \infty$.