Non-Markovianity in atom-surface dispersion forces

We discuss the failure of the Markov approximation in the description of atom-surface fluctuation-induced interactions, both in equilibrium (Casimir-Polder forces) and out of equilibrium (quantum friction). Using general theoretical arguments, we show that the Markov approximation can lead to erroneous predictions of such phenomena with regard to both strength and functional dependencies on system parameters. In particular, we show that the long-time power-law tails of two-time dipole correlations and their corresponding low-frequency behavior, neglected in the Markovian limit, affect the prediction of the force. Our findings highlight the importance of non-Markovian effects in dispersion interactions.

Figure 1

Schematic of an atom above a surface experiencing the Casimir-Polder force. The presence of the correlation tensor in the expression of the force Eq. (4) can be qualitatively understood as stemming from the interaction of the atom with its image within the material.

Figure 2

Non-Markovian correction to the Casimir-Polder force as a function of the atom-surface separation, estimated as the difference between the QRT and the FDT approach (see main text). The two curves correspond to different values of the atom-field coupling: The lower (blue) curve refers to ${}^{87}\mathrm{Rb}$, for which the ratio between the free-space linewidth and frequency is ${\gamma }_a^{\mathrm{free}}/{\omega }_a=5\times {10}^{-8}$ (${\omega }_a\approx 2.4\times {10}^{15}$ rad/s) [42]; the upper (red) curve is the result for ${\gamma }_a^{\mathrm{free}}/{\omega }_a=1/3$, typical of fullerene (for ${\mathrm{C}}_{60}{\omega }_a\approx 8.9\times {10}^{15}$ rad/s, see Ref. [41]). The behavior at large separations is associated with the oscillations of the decay rates in the retarded regime $z_{a}c/{\omega }_a\gg 1$. The planar surface is modeled as a metallic half space whose constituent material is described by the Drude model $\epsilon \left(\omega \right)=1-{\omega }_p^2{\left[\omega (\omega +i\mathrm{\Gamma })\right]}^{-1}$ with parameters that are typical for gold: ${\omega }_p=9$ eV and $\mathrm{\Gamma }/{\omega }_p=5\times {10}^{-3}$. The distance is measured in units of the plasma wavelength ${\lambda }_p=2\pi c/{\omega }_p$ ($\sim 140$ nm).

Figure 3

Comparison of the normalized power spectra $s\left(\omega \right)=\underline{S}·{(\mathbf{d}\mathbf{d}/\pi {\tilde{\omega }}_a)}^{-1}$ predicted by the FDT (blue line) and the QRT (red line) for the polarizability of Eq. (15) (dissipation is set to ${\gamma }_a/{\tilde{\omega }}_a={10}^{-2}$). While the two results are indistinguishable near resonance, they differ at low and negative frequencies. This is clearly visible in the inset, which provides a close up around $\omega =0$. While the power spectrum given by the FDT vanishes for $\omega \le 0$, that obtained from the QRT is nonzero.

Figure 4

Schematic of quantum friction on an atom moving at constant velocity parallel to a surface. As in the static case, the presence of the correlation tensor in the expression of the frictional force can be qualitatively understood as stemming from the interaction of the atom with its delayed image within the material.

Figure 5

Velocity dependence of the averaged second-order frictional force for a particle without intrinsic dissipation (e.g., an atom) moving above a metallic surface. The normalization is $F_0=-3\hslash {\omega }_{\mathrm{sp}}^5{\alpha }_0/\left(2\pi {\epsilon }_0{c}^4\right)$. The metal is described by the Drude model and results for various dissipation parameters are depicted: $\mathrm{\Gamma }=0$ [gray dashed, Eq. (35)], $\mathrm{\Gamma }/{\omega }_{\mathrm{sp}}={10}^{-3}$ (dark yellow), and $\mathrm{\Gamma }/{\omega }_{\mathrm{sp}}={10}^{-1}$ (purple). The other parameters are ${\omega }_a/{\omega }_{\mathrm{sp}}=0.2$ and $z_a{\omega }_{\mathrm{sp}}/c=0.05$, corresponding to the near-field regime, and $v_x/c=0.04$. The inset depicts the dependence of quantum friction for $\mathrm{\Gamma }=0$, Eq. (35), on the orientation of the vector of the dipole moment: The value $\theta =0$ corresponds to a dipole oriented normal to the surface; if tilted, the force has is largest value is found for a dipole in the $xz$ plane ($\varphi =0$).

Figure 6

Velocity dependence of the normalized averaged second-order quantum frictional force for a particle with intrinsic dissipation. The normalization is $F_0=-3\hslash {\omega }_{\mathrm{sp}}^5{\alpha }_0/\left(2\pi {\epsilon }_0{c}^4\right)$. The particle has an internal resonance frequency ${\omega }_a/{\omega }_{\mathrm{sp}}=0.2$ and moves at a distance $z_a{\omega }_{\mathrm{sp}}/c=0.05$ above the surface, corresponding to the near-field regime. Results for two different values of the intrinsic dissipation are displayed: $\gamma /{\omega }_{\mathrm{sp}}={10}^{-1}$ (dark yellow) and $\gamma /{\omega }_{\mathrm{sp}}={10}^{-3}$ (red). In both cases, the damping $\mathrm{\Gamma }$ is set to $\mathrm{\Gamma }/{\omega }_{\mathrm{sp}}={10}^{-1}$. The purple curve shows the case for $\gamma =0$ and $\mathrm{\Gamma }/{\omega }_{\mathrm{sp}}={10}^{-1}$. The black dotted curve represents the expression in Eq. (38). The thick gray dashed curve is Eq. (35).