Casimir-Lifshitz force for nonreciprocal media and applications to photonic topological insulators

Based on the theory of macroscopic quantum electrodynamics, we generalize the expression of the Casimir force for nonreciprocal media. The essential ingredient of this result is the Green's tensor between two nonreciprocal semi-infinite slabs, including a reflexion matrix with four coefficients that mixes optical polarizations. This Green's tensor does not obey Lorentz's reciprocity and thus violates time-reversal symmetry. The general result for the Casimir force is analyzed in the retarded and nonretarded limits, concentrating on the influences arising from reflections with or without change of polarization. In a second step, we apply our general result to a photonic topological insulator whose nonreciprocity stems from an anisotropic permittivity tensor, namely InSb. We show that there is a regime for the distance between the slabs where the magnitude of the Casimir force is tunable by an external magnetic field. Furthermore, the strength of this tuning depends on the orientation of the magnetic field with respect to the slab surfaces.

Figure 1

Sketch of the two semi-infinite half spaces with their boundaries at $z=0$ and $z=L$. The left half space is denoted by a minus sign and the right one by a plus sign. Moreover, the vectors $d{\mathbf{A}}^{-}$ and $d{\mathbf{A}}^{+}$ are orthogonal to the interfaces. The source point is located at ${\mathbf{r}}^{\prime }$ and the field point at $\mathbf{r}$. There are three reflections in total. The outgoing wave goes in the left direction and the incoming one goes to the right.

Figure 2

Numerically calculated Casimir force density for different static values of ${\varepsilon }_{ii}$ as a function of ${\varepsilon }_{xy}$ at a fixed gap distance of $L=10$ mm. The dashed lines correspond to ${\varepsilon }_{xx}=3$ and ${\varepsilon }_{zz}\to \infty$, the solid lines to ${\varepsilon }_{xx}={\varepsilon }_{zz}=3$ and the dotted ones to ${\varepsilon }_{xx}={\varepsilon }_{zz}=1$. Lines with (without) circles correspond to the case with $B^{+}=B^{-}$ ($B^{+}=-B^{-}$) in each case.

Figure 3

The Casimir force density $f$ has been calculated using Eq. (31) for the InSb model (38) as a function of the gap distance $L$ in the main plot without applied magnetic field (solid line) and for the case $B^{-}=-B^{+}=20$ T (dashed line). Additionally, $f$ was plotted only considering contributions from reflections with $f_{\text{off}}$ (solid line) or without $f_{\text{diag}}$ (dashed line) change of polarization in the inset for the case $B^{-}=-B^{+}=20$ T. Since $f_{\text{off}}<0$, the negative value $-f_{\text{off}}$ was plotted.

Figure 4

Numerically calculated ratio between the Casimir force density with and without external magnetic field $f\left(B=0\right)/f\left(B\right)$ as a function of the gap distance $L$ and for different magnetic fields $B=1$ T (triangles), 5 T (diamonds), 10 T (squares), 20 T (bullets). Furthermore, we distinguished between the cases with $B^{+}=B^{-}$ (dashed lines) and $B^{+}=-B^{-}$ (solid lines). For $B=20$ T we also plotted the numerical result in case of $r_{\mathrm{s},\mathrm{p}}, r_{\mathrm{p},\mathrm{s}}=0$ (dashed and dotted line) and so here the only nonvanishing contributions arise from $r_{\mathrm{s},\mathrm{s}}, r_{\mathrm{p},\mathrm{p}}$.

Figure 5

The Casimir force in the nonretarded limit approximation $f_{\mathrm{nret}}$ has been evaluated numerically for a fixed gap distance of $L=1$ nm as a function of the magnetic field $B=B^{\pm }$. Afterwards the ratio ${\tilde{f}}_{\mathrm{nret}}=f_{\mathrm{nret}}\left(B=0\right)/f_{\mathrm{nret}}\left(B\right)$ has been computed and plotted. As one can see here this reduction factor ${\tilde{f}}_{\mathrm{nret}}$ increases with increasing magnetic field until it saturates at around $B\cong 40$ T at ${\tilde{f}}_{\mathrm{nret}}\cong 2$.

Figure 6

The Casimir force density in the nonretarded limit approximation is plotted as a function of $\omega$ and $k^x$ for different magnetic fields $B=B^{\pm }$ and with $L=10$ nm, similar to Ref. [20]. Additionally, the dispersion relations for surface cavity modes are plotted with black solid lines in the $\omega -k^{\parallel }$ plane. Furthermore, one can find different regions I, II, III, in each figure separated by black dotted lines. In region I we have $\mathrm{Re}\left({\varepsilon }_{xx}\right),\mathrm{Re}\left({\varepsilon }_{zz}\right)<0$, and so this is the region with surface phonon and surface plasmon polaritons. In regions II and III there is $\mathrm{Re}\left({\varepsilon }_{xx}\right)\mathrm{Re}\left({\varepsilon }_{zz}\right)<0$ and we can find the hyperbolic modes HMI (II) and HMII (III), respectively.

Figure 7

Casimir force density in the nonretarded limit $f_{\mathrm{nret}}$ for a fixed gap distance $L=1$ nm as a function of the plasma frequency ${\omega }_{\mathrm{p}}$ for different fixed values of $B^{\pm }=0$ T (solid line), 1 T (dashed line), 5 T (dotted line), and 20 T (dashed and dotted line).

Figure 8

The different contributions from reflection with $\left(-f_{\mathrm{off}}\right)$ (solid line) or without $\left(f_{\mathrm{diag}}\right)$ (dashed line) a change of polarization at the interface are plotted separately for the parameters ${\omega }_e=\pm 7.7\times {10}^{12} 2\pi /\mathrm{s}$ and ${\omega }_0=\pm 1.3\times {10}^{10} 2\pi /\mathrm{s}$ and additionally ${\omega }_0{\omega }_e<0$ in both half spaces. Since $f_{\mathrm{off}}<0$ we plotted $-f_{\mathrm{off}}$. There is a region where $|f_{\mathrm{off}}|>|f_{\mathrm{diag}}|$ (see inset) and so that is where we expect to find a total repulsive Casimir force, cf. Fig. 9.

Figure 9

The Casimir force density of a PTI for the model similar to iron garnet as a function of the gap distance. We used in both half spaces ${\omega }_e=\pm 7.7\times {10}^{12} 2\pi /\mathrm{s}$ and ${\omega }_0=\pm 1.3\times {10}^{m_2} 2\pi /\mathrm{s}$ and different values for $m_2=10$ (dashed), 11 (dotted), and 12 (solid). Additionally, we distinguished between the two cases where first ${\omega }_0,{\omega }_e>0$ in both half spaces (without circles) and second ${\omega }_0, {\omega }_e<0$ in only one of the half spaces and ${\omega }_0,{\omega }_e>0$ in the other one (with circles).