Optical instabilities and spontaneous light emission by polarizable moving matter

One of the most extraordinary manifestations of the coupling of the optical field and matter is the emission of light by charged particles passing through a dielectric medium: the Vavilov-Cherenkov effect. In this article, it is theoretically predicted that a related phenomenon may be observed when neutral fast polarizable particles travel near a metal surface supporting surface plasmon polaritons. Rather dramatically, it is found that at some critical velocity, even if the initial optical field is vanishingly small, the system may become unstable and may start spontaneously emitting light such that in some initial time window the optical field grows exponentially with time.


I. Introduction
The interactions between light and matter are observed in many forms: the emission, absorption, and scattering of light, optical forces on nanoparticles, Raman scattering, just to name a few. In particular, the generation of light by charged particles passing either through a medium or near by a diffraction grating has been demonstrated by Vavilov & Cherenkov [1,2] and by Smith & Purcell [3], respectively, and has important applications in the detection of high-energy charged particles in astrophysics and particle physics.
These remarkable phenomena can be explained in the framework of classical electrodynamics because a modulated beam of moving charged particles corresponds to a time varying current, leading thus to the emission of light.
On the other hand, the emission of light due to fast changes in the geometry of electrically neutral macroscopic bodies (e.g. due to the accelerated motion of material boundaries) has also been extensively discussed in quantum physics in the context of the dynamical Casimir effect [4]- [9]. Moreover, several authors predicted the emission of radiation either by bodies in relative translational motion or by rotating objects [10]- [23].
These effects are understood as being intrinsic to quantum electrodynamics, and no classical analogue has been reported.
The playground for this work is the scenario wherein a neutral particle -with no net electric charge, e.g. an electric dipole -moves closely parallel to an uncharged metal surface. According to classical theory, the dipole motion should be totally unaffected by the presence of the metal surface, provided the electromagnetic field vanishes and the dipole is in its "ground state" (classical electric dipole moment is zero) so that there are -3-no charge oscillations. Indeed, a beam of neutral particles is supposedly uncoupled from the radiation field. Here, it is shown that, extraordinarily, if the velocity of the electric dipole is sufficiently large, a system instability may be developed so that the dipole oscillations and light emission can be triggered by vanishingly small optical noise. We note that related problems have been studied in the framework of quantum theory, in connection with the problem of quantum friction [10]- [23]. In particular, related optical instabilities have been recently linked to quantum friction in the case of nondispersive dielectric slabs [24][25][26] (see also Ref. [23]). Quite differently, here our analysis is fully classical, and both the electromagnetic field and the dynamics of the pertinent moving bodies are treated with classical theories.

II. Natural oscillations of an electric dipole above a moving half-space
Let us consider a vertical electric dipole (oriented along the z-direction) standing in freespace at a distance d from a planar thick metallic region [ Fig. 1a]. The relative velocity between the dipole and the metallic region is v, and is assumed time independent except if stated differently. As discussed later, having v independent of time may require some external action to counterbalance optically induced forces. For convenience, we take the dipole rest frame as the reference frame wherein all the physical quantities are defined.
To keep the formalism simple, it is supposed that the dipole can only vibrate along the zdirection (anisotropic particle), but as discussed ahead the theory can be readily extended to the general case where the dipole response is isotropic.
First, we characterize the fields radiated by the dipole when it oscillates with frequency  . The electromagnetic fields in the 0 z  region [ Fig. 1a] are the superposition of the -4-field radiated by the dipole in free-space and the field scattered by the moving metallic slab, inc s   E E E . The "incident" electric field is given by where ˆt Here we are interested in the natural oscillations of the system with 0 ext z E  . Clearly, the natural oscillations occur for frequencies  such that: Because of the radiation loss, manifested in the dynamics of the dipole oscillations by means of the Abraham-Lorentz self-force [27], the electric polarizability of a dipole oscillator is required to satisfy (in case there are no other mechanisms of loss) the Sipe-Kranendonk condition [28]   Lorentz-type dispersion model The sign of the term in the inner brackets is the same as the sign of

III. Compensation of the radiation loss
Surprisingly, it is shown next that when the relative velocity between the dipole and the moving medium is sufficiently large and is kept constant in the time window of interest, the system may become unstable and support exponentially growing oscillations For simplicity, first we use a non-relativistic approximation ( / 1 v c  ) to obtain When the effects of time retardation are neglected, Within this approximation, we obtain the following explicit formula for the interaction constant: In what follows, it is assumed that the metal permittivity has a Drude type dispersion   To unveil the conditions under which it is possible to have this optical instability, we derive an explicit formula for   int Im C in the limit of vanishing material loss ( 0    ).
In this limit one has Thus, from Eq. (13) and using it is easily found that in the lossless case: ). In is always positive. For velocities larger than the optimal value, the coupling between the dipole and the surface plasmons is less effective, and thus the dipole is required to move closer to the metal surface in order that the radiation loss can be over-compensated. Figure

IV. Conditions for an optical instability
As discussed next, the optical instabilities can be understood as being the result of the hybridization of the dipole resonance ( 0    ) and the SPPs supported by the silver slab ) in the metal co-moving frame. Here   is the Doppler shifted frequency, i.e. the frequency in the frame co-moving with the silver slab.
For simplicity, in this discussion a vanishing material loss is assumed ( 0    ). In the near-field approximation, the guided modes of the silver slab (i.e. the SPPs) occur for which correspond to the poles of the reflection coefficient when . It is this interaction between positive and negative frequencies that creates the opportunity to have the system instabilities and an exponentially growing oscillation.
More generally, one could consider the interaction of a dipole with a generic moving waveguide, for example a finite thickness dielectric slab. In this case, it is possible to write in the non-relativistic regime: . For a passive material the reflection coefficient is bound to the restriction: is the normalized wave admittance for p-polarized waves in the vacuum. In particular, the component of the time-averaged Poynting vector flowing towards the interface along the normal direction is For a passive material at rest one must impose that 0 av S  . Taking into account that p Y is pure imaginary when Notably, for a moving system the picture can change significantly. Indeed, from Eqs.   (13) and (14)]. It is wellknown that the poles of the reflection coefficient determine the dispersion of the guided Even though necessary, these conditions are insufficient to guarantee the emergence of an instability because the gain provided by an interaction between the dipole and a guided mode that satisfies (21) must also be sufficiently large to supplant all the loss channels From the selection rule (21), one also sees that for an interaction between positive and negative frequencies 0 x vk  . Moreover, for 0 y k  we see that: Interestingly, this threshold is the same as in the Cherenkov problem, which further confirms that the two phenomena are intrinsically related. Furthermore, similar to the Cherenkov problem, the instability threshold is determined by the phase velocity rather than by the group velocity.

V. Optically induced force
Evidently, to support growing oscillations the system must be pumped somehow.
Because there is no explicit optical pump the system is mechanically pumped. Indeed, in the previous calculations the relative velocity of the electric dipole and silver slab was assumed constant. As proven next, this is only possible if an external mechanical force that counteracts the optically induced forces is applied to the pertinent bodies. The optical force acting on the electric dipole is where 0 r represents the coordinates of the dipole center of mass [33,34]  In Fig. 3b and 3c we represent the local field (scattered by the moving silver slab) acting on a dipole driven at sp    . As seen, the local field consists of an SPP type wave dragged by the moving slab, such that for 0 x  the field is near zero. Clearly, to keep To estimate the strength of the involved forces, the electric dipole moment e p of the neutral particle at the initial time 0 t  is taken equal to the transition dipole moment e d of the Li I atom [31,32]. Moreover, the mass of the neutral particle is taken equal to 6 amu M  , consistent with the atomic mass of Li I. In this case, in the conditions of Fig.   3a the optical force acting on the particle at 0 t  is , 0.017  Fig. 3a, this force acts to pull the neutral particle towards the silver slab, and hence it is an attractive force analogous to the van der Waals-type forces arising from quantum and thermal fluctuations. The zcomponent of the force also exhibits an exponential growth. From a classical point of view, the emergence of this force is also rather surprising and is another signature of the optical instability. In a realistic experiment, this attractive force may limit the time that a small particle can travel above the metal surface without colliding with it.

VI. Which of the bodies emits the electromagnetic energy?
As previously discussed, the radiation stress necessarily does some work when the metal slab is sheared with respect to the dipole, and in a closed system this implies a transfer of kinetic energy to the radiation field. Very importantly, the source of the light generation is perceived differently by observers in different reference frames. If the initial electromagnetic field energy is vanishingly small in the reference frame co-moving with a given body (body A) the source of radiation must be the other body (body B). Indeed, only body B has kinetic energy to give away in the frame co-moving with body A. Thus, from a classical point of view, the dynamics of the process is perceived by an observer co-moving with a given object as radiation by the other object in relative motion. This is consistent with the fact that the optical instability results from the interaction of two oscillators (in case of Fig. 1a, the dipole resonator and a surface plasmon) that should be treated on the same footing, and which generate radiation only when they interact with each other.
In particular, from the point of view of the dipole the energy that pumps the system is radiated by the moving slab. A clue for where the energy comes from is given by Eq. (19), which gives the energy density flux ( av S ) created by an incident plane wave with transverse wave vector   , x y k k . For an evanescent wave with real-valued frequency  impinging on a moving slab (with velocity v in the considered reference frame) av S reduces to: The condition (18) ensures that 0 av S  when 0 v  . Surprisingly, one sees that when the velocity is sufficiently large so that Evidently, if the source is at rest in the frame co-moving with the medium, the condition (18) guarantees that 0 av S  for any incident wave. Yet, because an incident evanescent plane wave is also seen in another reference frame as an incident evanescent wave, it may seem paradoxical that av S can switch sign from one reference frame to another.
To make sense of this, let us consider a specific incident evanescent wave such 0 av S  in a reference frame where the slab has velocity 0 v  (for definiteness, we designate this frame as the laboratory frame). Next, we note (similar to what was already discussed in Sect. V) that the incident field can create an x-directed force that acts to change the velocity of the material slab. Clearly, in the reference frame co-moving with the slab the energy coming from the source can either be absorbed by the material medium or alternatively it can be used to increase the kinetic energy of the slab. Thus, av S is necessarily non-negative in the co-moving frame. However, in the laboratory frame The work done by the optical force is , is the power emitted per unit of area due to the conversion of kinetic energy into electromagnetic radiation: Note that 0 s P  in the co-moving frame, and hence s P is observer dependent. The conservation of energy requires that is the z-component of the Poynting vector at the interface. From Eqs. (18) and (24) it is seen that this condition is indeed satisfied: Thus, the finding that the sign of av S can be observer dependent is totally acceptable from a physical point of view, and does not violate energy conservation in any manner.
Moreover, this property is absolutely essential so that the interpretation of the phenomenon can be observer dependent, and that an instability is perceived by an observer co-moving with one the bodies as radiation originated in the other body. This is actually not any different from Cherenkov radiation: If an electric charge moves at a certain distance from a metal half-space (similar to Fig. 1a but for a charged particle) the generated light is perceived as being radiated by the charge in the medium co-moving frame. However, for another observer co-moving with the charge the generated light will be perceived as being emitted by the dipoles induced in the moving medium by the static electric field distribution created by the charge. Thus, the direction of energy flow must be observer dependent.

VII. Collective response of many electric dipoles
By considering many identical electric dipoles it may be possible to significantly enhance the Cherenkov-type instabilities, because the optical field emitted by a generic dipole may serve to drive the oscillations of other dipoles. To prove this we consider the scenario of Fig. 1b

VIII. Conclusion
In summary, we theoretically demonstrated that two neutral closely spaced polarizable bodies in relative motion can start spontaneously emitting light if their relative velocity is Crucially, the analysis of this article is completely classical, but the reported effects may lead to exciting developments in the framework of quantum electrodynamics, particularly in context of noncontact quantum friction [23][24][25][26]. Furthermore, the present theory (see also Ref. [24]) raises the question if there is anything specifically "quantum" in some phenomena involving the quantum vacuum [10]- [23], and suggests that "quantum friction" and related effects may eventually be explained with classical arguments with the additional ingredient of a spectrum of random electromagnetic radiation [36].
It is assumed that the interface is normal to the z-direction so that the transverse components of the electric field are tangential to the interface.
Suppose that the pertinent body moves with velocity v  v x in the laboratory frame. It is , which is defined in the same manner as   , , x y k k  R but for the frame co-moving with the body: All the quantities with the tilde hat are calculated in the co-moving frame. The electromagnetic fields in the two frames are related by [27]: Substituting this result into Eq. (A2), it is found that: are related to the corresponding parameters in the laboratory frame through the relativistic Doppler shift formulas [27]: Comparing Eqs. (A1) and (A5) we find that: For the case of an unbounded semi-infinite metal slab, it can be shown using vector transmission line theory that the matrix co R is given by: where c  Y is defined by where  and  are the relative permittivity and permeability, respectively, and  is the and (A8) one can easily determine the reflection matrix R in the laboratory frame.
In the non-relativistic limit 0  A in Eq. (A7), and hence we obtain: , ,  The lower region moves with a relative velocity v.