Fluctuating Forces Induced by Non Equilibrium and Coherent Light Flow
Casimir physics covers a wealth of phenomena where forces between macroscopic objects are induced by long range fluctuations M. Kardar, R. Golestanian (1999) of either classical or quantum origin. Fluctuations of the quantum electrodynamic (QED) vacuum epitomize this type of physics H. B. G. Casimir (1948), but such fluctuation induced forces (FIF) arise in a wide range of systems M. E. Fisher, P.-G. de Gennes (1978); A. Aminov, Y. Kafri, M. Kardar (2015); T. R. Kirkpatrick, J. M. Ortiz de Zárate, J. V. Sengers (2014); D. S. Dean, B.-S. Lu, A. C. Maggs, R. Podgornik (2016); R. Messina, D. A. R. Dalvit, P. A. Maia Neto, A. Lambrecht and S. Reynaud (2009).
Here we present an unexpected example of FIF caused by classical light propagating in a scattering medium. In weakly disordered media, light intensity has long ranged spatial fluctuations (speckle) associated to mesoscopic coherent effects resulting from elastic multiple scattering. These intensity fluctuations lead to measurable FIF on top of the disorder average radiation forces.
Quite remarkably, spatially coherent light fluctuations can be thoroughly described using a Langevin equation, where a properly tailored noise accounts for mesoscopic coherent effects. This non intuitive result allows to interpret mesoscopic FIF as resulting from a non equilibrium light flow. The strength of the FIF depends on a single dimensionless parameter – the analog of conductance in electronic systems, henceforth called conductance – which encapsulates both the geometry and the scattering properties of the random medium.
The scarcity of measurable and temperature independent non equilibrium phenomena makes the present proposal particularly relevant to experimental inspections. Indeed, since light induced fluctuating forces depend on the easily tunable parameter , coherent multiple light scattering offers setups where FIF are significantly enhanced compared to other known situations Lamoreaux (1997); A. Lambrecht, S. Reynaud (2000); J. N. Munday, F. Capasso, V. A. Parsegian (2009); G. Jourdan and A. Lambrecht and F. Comin and J. Chevrier (2009); Mohideen and Roy (1998); C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, C. Bechinger (2008). This should have interesting and a wide variety of applications such as new types of sensors in soft condensed matter, biophysics K. Bradonjić, J. D. Swain, A. Widom, Y. N. Srivastava (2009) and quantum technologies.
Precise Statement of Results
The mechanical actions of radiation forces have been well studied, e.g on dielectric bodies placed in vacuum. In random media, these forces result from multiple light scattering and they are expressed by phenomenological light currents obtained from an effective Langevin equation. On average, radiation forces depend on the transmission coefficient and the group velocity. But a yet unanticipated result is the existence of significant fluctuations of radiation forces around the average value, analogous to hydrodynamics fluctuating forces which occur in strongly non equilibrium systems A. Aminov, Y. Kafri, M. Kardar (2015). However, unlike systems hitherto considered, here the radiation forces fluctuations originate from underlying coherent mesoscopic effects associated to elastic multiple scattering.
To understand the origin of these FIF, we first note that, on average, light propagation in a scattering medium is well described by a diffusion approximation E. Akkermans, G. Montambaux (2007); Ishimaru (1978). Light intensity at a point results from the superposition of classical diffusion paths built out of the pairing of two phase coherent multiple scattering amplitudes of opposite phases. This phase information, hidden in the average light intensity, can be retrieved at a local crossing of two incoming diffusion paths which coherently exchange their phase coherent amplitudes so as to produce a new set of differently paired outgoing diffusion paths. The occurrence of these coherent exchanges (Fig.1.d) is controlled by a dimensionless parameter characteristic of the geometry and scattering properties of the medium 111This phenomenological picture for coherent mesoscopic effects is presented at an introductory level in E. Akkermans, G. Montambaux (2007), section 1.7.. Long range and incoherent diffusion paths propagate these local coherent effects over large distances. This mechanism, detailed hereafter, is at the origin of a wealth of coherent mesoscopic effects for wave (either electronic or photonic) propagation in random media which show up as spatially long range fluctuations of intensity.
These intensity fluctuations induce a new type of radiation forces (see Fig.1.a) fully understood and characterized by means of an effective Langevin description of the light flow, where coherent mesoscopic effects are the source of the noise. This approach is of particular interest since it maps the problem of coherent multiple light scattering onto an effective non equilibrium light flow characterized by two parameters only, the diffusion coefficient and the strength of the noise , otherwise related by a Einstein relation.
The Langevin Description
Consider the setup of Fig.1 where a random and -dimensional dielectric medium of volume is illuminated by a monochromatic scalar radiation 222Polarization effects are usually decoupled from disorder. For more elaborations, see E. Akkermans, G. Montambaux (2007). of wave-number incident along the direction of unit vector . Inside the medium, the amplitude of the radiation is solution of the scalar Helmholtz equation,
where denotes the fluctuation of the dielectric constant , is the average over disorder realizations and is the source of radiation of power . Besides the wave-number , the radiation in the medium is characterized by the elastic mean free path .
Multiple scattering solutions of the Helmholtz equation (1) in the presence of disorder are notoriously difficult to obtain. In the weak disorder limit , an equivalent description of the local radiation at a point and propagating along a direction is provided by the specific intensity , and the light current averaged over all directions E. Akkermans, G. Montambaux (2007); Ishimaru (1978) (see SM section 1.2), where is a conveniently defined group velocity. In this approach, the force exerted by light on an absorbing surface of normal vector , immersed inside the scattering medium, is
A Fick’s law of diffusion coefficient ,
relates the disorder averaged light current to the disorder and direction averaged intensity . The latter obeys a diffusion equation, whose solutions have the generic form
where is a dimensionless function determined by the geometry and boundary conditions and is a typical geometric size of the medium (see SM section 1.2). Inserting Eq.(3) into Eq.(2), allows to obtain the average radiation force . Its value, for an incident light beam perpendicular to a surface placed inside the medium at a distance from the incidence plane, is , where is the transmission coefficient 333Interestingly, the total average force on a closed surface surrounding the light source is equal to , i.e. independent of disorder. (see SM section 2).
All phase dependent effects have been washed out in the disorder average diffusive limit underlying Eq.(3). A well defined semi-classical description enables to include coherent effects in a systematic way. It starts by noting (see Fig.1.c) that each diffusive trajectory is built from the pairing of two identical but time reversed multiple scattering amplitudes obtained from scattering solutions of Eq.(1). By construction, these two amplitudes have opposite phases so that the resulting diffusive trajectory is phase independent. Unpairing these two sequences gives access to the underlying phase carried by each multiple scattering amplitude and thereby to phase coherent corrections. The aforementioned description makes profit of this remark to evaluate phase coherent corrections (see Fig.1.d). At a local crossing, two diffusive trajectories mutually exchange their phase so as to form two new phase independent diffusive trajectories. This local crossing – or quantum crossing – is a phase dependent correction propagated over long distances by means of diffusive trajectories Note1 (). The occurrence of a quantum crossing in a disordered medium of volume is controlled by a single dimensionless parameter, its conductance,
which depends on the geometry and on scattering properties of the medium. From now on and without loosing in generality, we consider the three dimensional () setup displayed in Fig.1. In the weak disorder limit , the conductance and small coherent corrections generated by quantum crossings show up as powers of . This scheme allows to expand spatial correlations of the fluctuating light intensity as
The first contribution (see Eq.(S47)) is short ranged and independent of . The two other contributions are long ranged, and respectively proportional to and . All three terms contribute to specific features of interference speckle patterns Goodman (2000), and have been measured in weakly disordered electronic and photonic media E. Akkermans, G. Montambaux (2007); M. Kaveh, M. Rosenbluh, I. Freund (1987); F. Scheffold, G. Maret (1998); F. Scheffold, W. Hartl, G. Maret, E. Matijevic (1997); J. F. de Boer, M. P. van Albada, A. Lagendijk (1992); M. J. Stephen, G. Cwilich (1987).
This expansion can be obtained in a different but completely equivalent and elegant way by noting that quantum crossings occur at lengths of order , smaller than the elastic mean free path . This allows to separate large scale incoherent diffusive physics from small scale, coherent and phase preserving quantum crossings. This partition is described by a Langevin equation,
which extends the Fick’s law, Eq.(3), to the fluctuating, not disorder averaged quantities and , by adding a zero average noise defined by the vector . This picture, originally presented in B. Z. Spivak, A. Yu. Zjuzin (1987), allows to reproduce the expansion of Eq.(6) by systematically including quantum crossings contributions in . To lowest order in (SM section 3),
where . We can rewrite the noise term under the form, , where 444We assume a gaussian noise in Eq.(8). It is justified since the FIF do not depend on higher moments., and a strength,
which depends quadratically on the average diffusive radiation intensity 555Note that the Langevin approach is valid for a noise whose amplitude is small, i.e. when scales down to zero with the system size. In rescaled units, , , Eq.(7) becomes: , so that the Langevin approach is justified for (see SM section 4)..
This effective Langevin description, based on the two parameters and , provides a complete hydrodynamic description of the coherent light flow in the random medium. Moreover, it is appealing since its specific dependence upon a constant and a quadratic , immediately draws a relation with the Kipnis-Marchioro-Presutti (KMP) process – a heat transfer model for boundary driven one dimensional chains of mechanically uncoupled oscillators strongly out of equilibrium C. Kipnis, C. Marchioro, E. Presutti (1982); L. Bertini, D. Gabrielli, J. L. Lebowitz (2005), well described by the macroscopic fluctuation theory L. Bertini, A. De Sole, D. Gabrielli et al. (2015). A correspondence with this process is obtained by formally identifying the radiation intensity to the energy density, and to the heat flow 666The Langevin equation (7) is time-independent, so that the correspondence is obtained by integrating the KMP Langevin equation over short time scales up to the elastic mean free time (see SM section 4).. Despite this formal correspondence, it is essential to note that the physical source of non equilibrium is very different in the two cases. While in the KMP model, energy density fluctuations result from thermal effects due to the coupling to two reservoirs at distinct temperatures, intensity fluctuations of the light flow result solely from the illumination of the random scattering medium.
A general Einstein relation exists which relates the parameters and characteristic of the hydrodynamic regime of strongly non equilibrium systems. It is given by , where is the static compressibility L. Bertini, D. Gabrielli, J. L. Lebowitz (2005); Spohn (1991). For the coherent light flow,
which from Eq.(9), satisfies the Einstein relation (SM section 4).
Fluctuation Induced Forces
We are now in a position to calculate the radiation force which includes on top of its average , a fluctuating (FIF) part induced by intensity fluctuations. In the geometry of Fig.1, a dielectric plate, or membrane, of surface along , is inserted in the scattering medium so as to cancel by symmetry the average force . The fluctuating part is readily obtained by substituting Eq.(7) into Eq.(2) together with Eq.(3) and it is given by
where is the counterpart of the corresponding term in Eq.(6) and results from the noise term. The contribution is always negligible compared to , as can easily be seen by considering the corresponding fluctuating forces on the faces of a cubic geometry with and without inner plate. The expression of together with Eqs.(4,8), implies immediately that , hence is negligible. The term induced by is of order and therefore also negligible. Finally, the behaviour of is readily obtained from Eqs.(4,8,10), namely where is a dimensionless number characteristic of the system geometry. Then, as can be anticipated from Eq.(6), behaves like and it is proportional to (SM section 5.1), so that finally the fluctuating force has the general form,
This rather simple expression constitutes a central result of this work. It states that the fluctuating forces induced by coherent mesoscopic effects, besides their dependence upon the power of the incoming light beam, are driven by the dimensionless conductance of the system. This conductance takes the simple form given in Eq.(5) for the geometry of the empty hypercube . For more involved geometries like in Fig.1, the conductance takes in , the general form where the length depends on the geometry of the scattering medium (see SM section 5.2 for examples) 777See section 2 in E. Akkermans, G. Montambaux (2007).. The two dimensionless numbers and in Eq.(12) depend on the shape of the system – but not on its volume – and on boundary conditions imposed on the average intensity in Eq.(4) (see SM section 5.1 for a detailed proof).
Playing with geometry to monitor forces
Fluctuation induced forces Eq.(12) depend on the geometry and boundary conditions, which opens a wide choice of parameters to control and amplify the forces. In the geometry of Fig.1, the highest values of the dimensionless s are obtained using reflecting cavity edges in the direction of the light beam and absorbing lateral edges. Interestingly, or can be independently enhanced by an appropriate choice of boundary conditions on the plate (SM section 5.2). On an absorbing plate where (Fig.2.a), only contributes with a maximum for an optimal value of (SM section 5.1). Alternatively, inserting a reflective plate with selects and leads to FIF with a power law dependence with (Fig.2.b). This limiting case has an interesting consequence since a measurement of finite FIF on a reflective plate demonstrates the existence of the noise term in the Langevin description of mesoscopic coherent effects (SM section 5.2).
Numerous efforts have been recently made to develop high sensitivity cantilevers able to measure forces of weak amplitude R. Castillo-Garza and Mohideen (2009). We propose to observe mesoscopic FIF using an atomic force microscope, in a setup similar to J. N. Munday, F. Capasso, V. A. Parsegian (2009) where Casimir-Lifshitz forces of a few piconewtons have been measured between a gold plate and a sphere coated with gold in a liquid. Replacing the liquid by a weakly scattering medium and using square plates of size mm – the typical size of the sphere used in J. N. Munday, F. Capasso, V. A. Parsegian (2009) – and illuminating the medium with a light beam of intensity W , we expect light FIF of amplitude up to a few hundreds of piconewtons, i.e. strong enough to be detected. These results are summarized in Table 1.
Physical aspects of diffusive light propagation either incoherent or coherent have already been hitherto studied in the literature. For electronic waves, the focus is mainly on transport properties, better accessible in mesoscopic devices and which stand as a favorite candidate to observe the elusive Anderson localization transition for large enough disorder. For radiation and other classical waves, transmission properties and long range correlations either spatial or spectral, have been also extensively studied. Despite these thorough investigations, mechanical effects resulting from coherent mesoscopic effects of diffusive light presented here have never been envisaged. They open a new and alternative approach to the field. From a fundamental viewpoint, the existence of fluctuation induced forces easily and solely monitored by the dimensionless conductance Eq.(5) has a threefold interest. First, the analogy here unveiled, between long range induced forces in a coherent mesoscopic light flow and in non equilibrium systems, should arouse experimental attention to observe such forces in the realm of radiation flow in Casimir physics. Second, coherent mechanical forces are sensitive to the disorder strength through the conductance and hence can be used as a new effective probe to study the existence and criticality of Anderson localization transition. Third, potential applications of coherent mechanical forces induced by a coherent diffusive radiation flow are diverse and promising: in addition to transmission measurements extensively used, they provide a new type of mechanical and sensitive sensors at submicronic scale rather easy to implement and useful in soft condensed matter, biophysics K. Bradonjić, J. D. Swain, A. Widom, Y. N. Srivastava (2009), nanoelectromechanical (NEMS) and quantum technologies T. J. Kippenberg and K. J. Vahala (2007); O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann (2006). Besides, we wish to highlight that the mapping we have presented between coherent light flow and out of equilibrium hydrodynamics is easily generalisable to a large class of quantum or classical mesoscopic effects, e.g in nanoelectronics and superconductivity L. Henriet, A. N. Jordan and K. Le Hur (2015). A clear asset of this type of approach is in its dependence upon two parameters only, thus making it a candidate to efficient machine learning algorithms.
This work was supported by the Israel Science Foundation Grant No. 924/09. We are grateful to Ohad Shpielberg for discussions, Boris Timchenko, Marc Soret and Igor Khmelnitsky for a critical reading of the manuscript and Yaroslav Don for his help in the preparation of the manuscript.
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