Casimir pressure in a multilayer system with a fixed total length

We consider Casimir pressure on a slab in a configuration consisting of various dielectrics with planar symmetry. In such configurations, one usually calculates the Casimir pressure (force) on a particular slab assuming that lengths of all other slabs remain unchanged. Alternatively, one can consider a multilayer system with a fixed total length. With this restriction only, the length of each slab can eventually be changed under the Casimir pressure that will try to minimize the total Casimir energy of the system. Here we calculate the Casimir pressure on the slab in such a “constrained” configuration and compare the results with the standard approach. It turns out that, by applying different boundary conditions, one can obtain significantly different Casimir pressures on the same object. In particular, when the thicknesses of the slab and surrounding layers are on the nanometer scale, the Casimir pressure on the slab can change from strongly squeezing in the case of fixed thicknesses of surrounding slabs to strongly relaxing in the case when only the length of the total system remains fixed.

Figure 1

Geometry of the model.

Figure 2

Casimir pressures $P_C^j$, $j=(0,1,2)$, (in units of $P_P$) as a function of the vacuum layer thicknesses $d_0=d_2$ (in units of ${\lambda }_P$), for different thicknesses of the metallic slab: $d_1=0.001$ ${\lambda }_P$ (dash-dotted lines), $d_1=0.01$ ${\lambda }_P$ (solid lines), and $d_1=0.1$ ${\lambda }_P$ (dashed line). Dotted lines: SP contribution $P_S^j$ to the Casimir pressures for $d_1=0.01$ ${\lambda }_P$.

Figure 3

Casimir pressures as a function of the thickness of the metallic slab $d_1$, for different vacuum layer thicknesses ($d_0=d_2$): $d_2=0.001$ ${\lambda }_P$ (dash-dotted lines), $d_2=0.01$ ${\lambda }_P$ (solid lines), and $d_2=0.1$ ${\lambda }_P$ (dashed line). Dotted lines: SP contributions to the total Casimir pressures for $d_2=0.01$ ${\lambda }_P$.

Figure 4

Casimir pressures $P_C^0$, $P_C^1$, and $P_C^2$ and the total Casimir pressure $P_C^{T1}=P_C^1-P_C^2$ on a thin Na slab $(d_1=0.01$ ${\lambda }_P)$ as a function of $d_0=d_2$. The outer layers are taken to be ideal mirrors (solid lines), Al (dashed lines), and NaCl (dash-dotted lines). The dotted line represents the SP contribution $P_S^{T1}=P_S^1-P_S^2$ to the total Casimir pressure in the ideal mirrors case.

Figure 5

Casimir pressures $P_C^0$, $P_C^1$, and $P_C^2$ (dashed lines) and the resulting Casimir pressures $P_C^1-P_C^0$ and $P_C^1-P_C^2$ (dashed-dotted lines) on a $d_1=0.01$ ${\lambda }_P$ thick metallic slab as a function of the vacuum layer thickness $d_2$. The total Casimir pressure $P_C^{T1}$ (solid line) has a discontinuity at $d_2^c=0.0103$ ${\lambda }_P$. The cavity walls are ideal mirrors, with the vacuum separations $(d_0+d_2)=0.04$ ${\lambda }_P$.

Figure 6

Casimir pressures as in Fig. 5, but for ten times larger characteristic thicknesses [$d_1=0.1$ ${\lambda }_P$, $(d_0+d_2)=0.4$ ${\lambda }_P$].

Figure 7

Casimir pressures $P_C^1$ and $P_C^0=P_C^2$ as a function of the thickness of the metallic slabs $d_0=d_2$, for different vacuum layer thicknesses: $d_1=0.001$ ${\lambda }_P$ (dash-dotted lines), $d_1=0.002$ ${\lambda }_P$ (dotted lines), $d_1=0.01$ ${\lambda }_P$ (solid lines), and $d_1=0.1$ ${\lambda }_P$ (dashed lines). The cavity walls are ideal mirrors. Inset: $P_C^1$ and $P_C^2$ on larger scale, for $d_1=0.001$ ${\lambda }_P$.

Figure 8

The same parameters as in Fig. 7, but with Na slabs within the NaCl cavity walls. Inset: $P_C^1$ and $P_C^2$ on a smaller scale, for $d_1=0.01$ ${\lambda }_P$.

Figure 9

Casimir pressures $P_C^0$, $P_C^1$, and $P_C^2$ and the total Casimir pressure $P_C^{T1}=P_C^1-P_C^2$ acting on a thin $(d_1=0.01$ ${\lambda }_P)$ vacuum layer in between the Na slabs, as a function of the slab thicknesses $d_0=d_2$. The outer layers are taken to be Al (dashed lines), Cs (dotted lines), NaCl (dash-dotted lines), and Na (solid lines).

Figure 10

Casimir pressures $P_C^0$, $P_C^1$, and $P_C^2$ (dashed lines) and the total Casimir pressure $P_C^{T1}$ (solid lines) acting on the thin vacuum layer $(d_1=0.01$ ${\lambda }_P)$ in between the Al ($j=0$) and Cs ($j=2$) slabs, as a function of the slab thicknesses $d_0=d_2$. The outer medium is Na.