Theory of Quantum Friction

Here, we develop a comprehensive quantum theory for the phenomenon of quantum friction. Based on a theory of macroscopic quantum electrodynamics for unstable systems, we calculate the quantum expectation of the friction force, and link the friction effect to the emergence of system instabilities related to the Cherenkov effect. These instabilities may occur due to the hybridization of particular guided modes supported by the individual moving bodies, and selection rules for the interacting modes are derived. It is proven that the quantum friction effect can take place even when the interacting bodies are lossless and made of nondispersive dielectrics.


I. Introduction
Quantum friction is a theory that predicts that two uncharged polarizable bodies moving relative to each other experience a force of quantum origin that tends to work against the relative motion [1][2][3][4][5][6][7][8]. This effect is predicted to take place even a zero temperature and when the surfaces of the moving bodies are flat and perfectly smooth (the materials are regarded continuous media). A physical picture is that the electric dipoles created by the quantum fluctuations in one of the surfaces induce image electric dipoles on the other surface, which, when the bodies are in relative parallel motion, lag behind and originate a van-der-Waals type attraction [1]. The quantum friction theory is not consensual and has been recently debated [4,9,10]. Quantum friction has also been studied in rotating dielectric bodies [11][12][13].
For many authors, quantum friction is understood as a purely quantum effect with no classical analogue. Recently, it was shown in Ref. [8] that for the case of sliding monoatomic surfaces the effect of friction is associated with electromagnetic instabilities in moving media that can lead to the creation of polaritons. Very interestingly, these electromagnetic instabilities are partly connected to the Cherenkov [14] and Smith-Purcell effects [15], and can be predicted by classical electrodynamics. Related electromagnetic instabilities have been discussed in the context of plasma physics, with application in the design of terahertz traveling-wave oscillators and amplifiers [16].
These electromagnetic instabilities are manifested in the fact that the system may support natural modes of oscillation that grow exponentially with time [8,16,17], even in presence of material loss [18]. In this article, we establish the definite and missing link between this classical effect and quantum friction.
Most of the available theories of quantum friction are based either on semi-classical arguments or on first order perturbation quantum theory. In Ref. [17], we developed (in the framework of macroscopic quantum electrodynamics) a theory for the quantization of the electromagnetic field in moving media systems with electromagnetic instabilities.
Using this formalism, here we derive the quantum friction force from first principles at zero temperature, and prove that it has a dynamical character, i.e. the expectation of the friction force varies with time. We prove that in the "pseudo-ground" state of the system [17], i.e. in the state wherein the oscillations of the quantum fields in the two moving bodies are minimal, the expectation of the friction force vanishes. However, as time passes the friction force builds up exponentially, as long as the velocity of the bodies is enforced to be constant through the application of an external force. Interestingly, we establish a precise connection between our theory and the semiclassical theory of Pendry [1,4]. We prove that Pendry's friction force corresponds to our dynamic friction force calculated at the time instant wherein the first "excitation" is generated.
The usual explanation found in the literature for quantum friction is related to material loss, such that the frictional work done on a given body is dissipated in the electrical resistance of the dielectric [3][4][5]. For example, in Ref. [8] it was found that the friction force vanishes in the lossless limit wherein the material responds instantaneously to the local fields. However, the analysis of Ref. [8] ignores the retardation effects due to the finite speed of light ( c   ), and thus in the absence of material dispersion the interaction between different electric dipoles is effectively instantaneous, and, moreover, -4-it is impossible to surpass the Cherenkov critical velocity when c   . Here, we prove that when wave retardation is properly taken into account it is possible to have a frictional force even when the local material response is instantaneous. This demonstrates that the friction effect does not require material loss. Thus, surprisingly, the friction force can be nonzero even when the interacting bodies are made of nondispersive lossless dielectrics. In a recent work [19], Maghrebi et al, have independently demonstrated (based on a scalar field theory) a connection between noncontact friction and Cherenkov radiation, consistent with our studies [8,17] (see also Ref. [20]). The emergence of electromagnetic instabilities above the threshold velocity for quantum Cherenkov emission was however not discussed by the authors of Ref. [19].
The article is organized as follows. In Sect. II, we review and extend the formalism developed in Ref. [17], which is the basis of our theory. In Sect. III, the friction force quantum operator is derived. In Sect. IV, the selection rules for guided modes that originate system instabilities are obtained. These selection rules complement the findings of Ref. [17]. In Sect. V we compute the quantum expectation of the friction force, and Sect. VI reports several numerical examples and an explicit comparison with the theory of Pendry. The conclusion is drawn in Sect. VII.

A. Material bodies coupled by the electromagnetic field
We are interested in the dynamics of a set of rigid lossless non-magnetic material bodies (i=1,2,…) coupled by the electromagnetic field. Let i v be the velocity of the i-th body center of mass. Consistent with our previous work [17], it is shown in Appendix A that provided / 1 i v c  the total energy ( tot H ) of the system can be written as, ) is not purely electromagnetic and includes also part of the energy stored in matter (e.g. the energy associated with dipole vibrations and part of the energy associated with the translational motion) [17]. It is proven in Appendix A that our system satisfies exactly the following conservation law: where ext j is an hypothetical external electric current density (in this work 0 ext  j ), and

B. Material bodies with time independent velocities
Next, we consider a system of polarizable non-dispersive moving bodies invariant to translations along the x and y directions (Fig. 1a). It is supposed that the relevant bodies between two moving bodies such that the wave energies stored in each of the bodies have opposite signs can originate system instabilities so that the electromagnetic field may support natural modes of oscillation with i       complex valued. In particular, when 0   the electromagnetic field oscillations may grow exponentially in time, as long as the velocity of the moving bodies is kept constant [17]. These instabilities take place even though there is no explicit source of excitation and in presence of strong material loss [18]. v v  in this example. For simplicity, we consider only the contribution of a pair of oscillators associated with the complex valued frequencies -10-

C. The force acting on a moving slab
As already discussed in Ref. [17], the wave instabilities imply the emergence of a friction force associated with the radiation drag whose effect is to act against the relative motion of the bodies. For closed systems ( 0 ext  j and , 0 ext tot i  F ), this feedback mechanism results in a decrease of the relative velocity of the bodies, and this ultimately prevents the continued exponential growth of the fields. From Eq. (4) one sees that in the absence of an external force the time rate of change of total momentum (i.e. the sum of the matter and electromagnetic momenta) enclosed in the i-th slab is , / wv i dp dt . Thus, , / wv i dp dt is a stress associated with the wave flow: It will be seen that in the quantum vacuum this stress acts against the relative motion, and will be responsible by a friction force. Thus, in the absence of an external force the velocity of the moving bodies typically changes with time.
Thus, the velocities can remain constant  as will be assumed in this article  only at the expense of applying an external action that counterbalances the friction force. It is seen from Eq. (4) that the external force required to maintain the velocity of the is time independent, the total energy (including the degrees of freedom associated with the translational motion of the system) increases with time. Moreover, even though the wave energy associated with a specific body can be negative (when M is indefinite) it turns out that the total energy stored in the body is always positive (see Eq. (A10) of Appendix A) [17].

III. Friction force operator
In order to determine the friction force we expand the electromagnetic field in terms of natural modes of oscillation. It was proven in Ref. [17], under the assumption that the velocities are enforced to be constant, that the eigenmodes of the system are either associated with real-valued frequencies ( n n   , , , , where . | . denotes the indefinite inner product of Eq. (9). The electromagnetic modes nk f and nk e are related by n xx yy zz the transformation matrix associated with the 180º rotation around the z-axis. The electromagnetic field in the cavity can be expanded as The coefficients of the expansion are ˆn c k , ˆn  k , and ˆn  k . The hat "^" indicates that in the framework of a quantum theory the pertinent symbol should be understood as an operator. In the framework of a classical theory, the coefficients ˆn does not vanish in presence of system instabilities associated with complex-valued frequencies of oscillation. It should be emphasized that this conclusion is valid for both classical and quantum systems, and thus the friction force has a classical counterpart.
In quantum theory the field amplitudes  , and  associated with a pair of complexvalued frequencies are written in terms of annihilation and creation operators , Substituting this result in Eq. (15) and summing over all the complex-valued oscillators (crossed terms associated with possible contributions from oscillators nk and mk , with n m  , are neglected; the quantum expectation of ˆt ot i F is independent of the crossed terms) it is found that (see also Eq. (B10)): . (17) The friction force acting on the matter enclosed by the i-th slab is given by a similar

IV. Selection rules
As demonstrated in the previous section, the friction force is a consequence of system instabilities manifested in the form of natural modes of oscillation with complex valued frequencies. In our previous work [17], we have shown using perturbation theory that these modes are the result of the interaction of guided waves supported by the moving bodies. In Appendix C, considering the limit of a weak interaction and that / 1 (C4) and (C5)] that the hybridization of the two guided modes results in a natural mode with a complex valued frequency provided the following selection rules are satisfied: Note that it is implicit that   x y Thus, at least one of the bodies moves with a velocity larger than the corresponding Cherenkov threshold in the laboratory frame.
To further develop these ideas, we consider the particular case wherein the dielectric slabs are identical. For each slab, the guided modes are characterized in co-moving frame by the dispersion branches The index n labels the guided modes branches. The simplest possibility (but not the only one) to satisfy the selection rules (18) is to consider the interaction of modes associated with the same n such that  To illustrate the discussion, we show in Fig. 2a . This is a general consequence of the selection rule (18a) when 1 We verified (not shown here) that if the effect of loss is considered in the material response it is still possible to have natural modes associated with growing oscillations (see also Ref. [18]). This result is also evident because of analytic continuation arguments. We also verified that relativistic corrections result in a small shift of the modal diagrams. -19-For future reference, it is mentioned that the sign of the pseudo-momentum associated with a system instability is such that in the i-th slab The proof is given in Appendix D.

V. Quantum expectation of the friction force
In the framework of a quantum theory, a generic state that the lifetime of the generated excitations is very short (e.g. they are quickly absorbed due to loss in the system) as compared to   1/ 2 n  k , because the system is assumed to be always in the ground state. Quite differently, in our approach the dielectrics are lossless and hence the generated excitations stay in the system and promote new excitations making , g n R k effectively time dependent. This is why in our theory the force grows exponentially, whereas in Pendry's theory the force is constant.
Since the average time to generate an excitation is of the order of   where , 1,2 i j  and i j  . This confirms that the friction force acts against the relative motion of the two slabs. In Appendix C, we give a detailed proof that 1, tot i g F exactly coincides with the calculation of Pendry in the limit of a weak interaction between the moving bodies [1,4]. This validates the previous discussion, and demonstrates that the semiclassical theory of Pendry can be recovered from our dynamic quantum theory.
It is interesting to mention that the absolute value of A the transverse area of the slabs in the xoy plane), we find that:

VI. Numerical Examples
To illustrate the application of the theory, we computed ˆt ot i F for the geometry of Fig. 1a supposing  In Fig. 3 we depict the calculated normalized friction force as a function of the It is striking from Fig. 3  . This is to some extent surprising because the p-polarized modes have a dispersion branch with no frequency cut-off, unlike the s-polarized modes which can only propagate above a certain threshold frequency (see Fig. 2a). A consequence of this is that for a fixed 2 1 2 v v c  the p-polarized modes that contribute to the force have smaller [4]. In Figure 4 we plot the friction force variance computed using the theory of this work

VII. Conclusion
A first principles derivation of the quantum friction force was presented. It was proven that the expectation of the quantum friction force vanishes at the initial time instant for a system of non-dispersive moving bodies prepared in the "pseudo-ground" state.
However, as time passes the expectation of the quantum friction force exponentially grows, as long as the change in the velocity of the moving bodies is insignificant. We calculated the quantum friction force at the time instant corresponding to the generation of the first excitation for each pair of unstable oscillators, and demonstrated that in the limit of a weak interaction it is coincident with the semiclassical result of Pendry [1,4].
The velocity threshold above which quantum friction can take place was derived, and the effect of quantum friction was linked to system instabilities that may occur when at least one of the dielectric bodies has velocity larger than the Cherenkov emission threshold.
These instabilities are due to the hybridization of guided modes supported by the individual bodies, and take place when the guided modes satisfy certain selection rules.
We numerically estimated the quantum friction when two non-magnetic grounded dielectric slabs are in relative motion, and demonstrated that, surprisingly, in the nonrelativistic limit the quantum friction is mainly determined by s-polarized waves. It is relevant to mention that the friction mechanism described in this article cannot be used to extract an infinite amount of energy from the system or achieve a "perpetuum mobile" [10]. Indeed, the "perpetuum mobile" argument of Ref. [10] relies on the hypothesis that we can have an ordinary passive magneto-electric material with the same constitutive relations as those of a moving slab. Clearly, in presence of wave instabilities this is impossible unless the magneto-electric material is active.
Acknowledgements: This work is supported in part by Fundação para a Ciência e a Tecnologia grant number PTDC/EEI-TEL/2764/2012.

Appendix A:
In this Appendix, we study the electrodynamics of moving rigid bodies in the nonrelativistic limit. The analysis is in part related to Ref. [30], which investigates the same problem but for moving electric dipoles, while here we consider the continuous limit.
-30-As a starting point, we note that the electromagnetic field dynamics is determined by: where , 0 co e i   is the electric susceptibility of the moving body in its rest frame. Note that i   E v B is the electric field in the co-moving frame.
where n is the outward unit vector normal to the surface. But straightforward manipulations of Eqs. (A2) and (A3) show that the right hand side of the previous formula equals , , Therefore, we demonstrated the energy conservation law: In case V is taken as all space, this result reduces to Eq. (3), being tot H defined as in Eq.
(1) the total energy of the system. It interesting to note that the energy stored in the i-th body can be identified with: Using Eqs. (A3) and (A4) this can also be written as:  For complex-valued poles the imaginary part  satisfies: