Experimental observation of effective gravity and two times physics in ferrofluid-based hyperbolic metamaterials

Recently it was proposed that extraordinary light waves in hyperbolic metamaterials exhibit two times physics behavior (Phys. Rev. Lett. 105, 067402, 2010). We report experimental observation of this effect via investigation of gravity-like nonlinear optics of iron/cobalt-based ferrofluid hyperbolic metamaterials. In addition to conventional temporal coordinate, the spatial coordinate oriented along the optical axis of the metamaterial also exhibits timelike character, which leads to very unusual two times physics behavior in these systems.

dielectric host of the metamaterial [8]. Experimental observation of the effective gravity in such a system should enable observation of the emergence of the gravitational arrow of time along the z direction, which is predicted to occur even at the classical gravity level [9]. Such an observation would further extend experiments on the emergence of the arrow of time in planar hyperbolic metamaterials reported in [7].
On the other hand, the model of 2+1 metamaterial Minkowski spacetime described above may itself evolve in our own physical time, which means that a theoretically proposed metamaterial-based 2T physics system [1] will be realized and observed in the experiment. In this paper we report an experimental study of gravitylike nonlinear optical effects in a three-dimensional self-assembled hyperbolic metamaterials based on iron-cobalt ferrofluid subjected to external magnetic field.
Magnetic interaction of the iron-cobalt nanoparticles in the ferrofluid is rather weak, so in the absence of external magnetic field the nanoparticles are randomly distributed within the fluid, as illustrated in Fig. 1a. On the other hand, application of a modest external magnetic field (of the order of 300 Gauss) leads to formation of nanocolumns aligned along the external field (see Fig. 1b), thus leading to formation of a selfassembled 3D metamaterial, which exhibit hyperbolic behaviour in the long wavelength infrared (LWIR) range [10]. Similar to other common fluids, kerosene (used as a host fluid for the cobalt nanoparticles) exhibits negative self-defocusing Kerr effect in the strong optical field of a CO 2 laser operating at the 10.6 m wavelength. Therefore, in agreement with theoretical predictions [8], the fabricated ferrofluid indeed exhibited pronounced gravity-like self-focusing effects, leading to emergence of the gravitational arrow of time [9] oriented along the optical axis z of the metamaterial. On the other hand, absorption of the LWIR light in the metamaterial leads to gradual heating of the system, thus leading to irreversible evolution of the system as a function of conventional physical time. A detailed account of these experiments, which realize the theoretically proposed 2T physics scheme [1] is reported below. 4 Schematic diagram of our experimental setup is shown in Fig. 1c. A sample of ferrofluid is contained between two NaCl windows separated by a thin (100 m) spacer within an optical cuvette and secured vertically to the optical table. A horizontally polarized CO 2 laser beam is then passed through the sample and a LWIR camera (FLIR Systems) is used to image the beam shape after its passage through the ferrofluid (it was verified that the original beam shape of the laser may be well described by a simple Gaussian profile -see inset in Fig. 1c. To see the effect of a DC magnetic field on the beam shape, which is associated with formation of a self-assembled hyperbolic metamaterial inside the ferrofluid, a large permanent magnet is placed near the cuvette (above or to the side), producing a magnetic field aligned almost vertically or almost horizontally (but not parallel to the cuvette). As a result, the laser light passing through the metamaterial becomes either predominantly ordinary (E field of the light field oriented perpendicular to the optical axis of the metamaterial), or predominantly extraordinary (E field having non-zero component parallel to the optical axis). Temporal evolution of the beam shape was studied as a function of magnetic field direction and laser intensity.
The iron-cobalt ferrofluid used in these experiments was similar to the cobalt ferrofluid used in [10], since the same carrier fluid (kerosene) was used in both cases and the optical properties of cobalt and iron are similar in the LWIR range [11]. The ferrofluid fabrication will be described elsewhere. photons. In order for the effective gravitational constant to be positive, negative selfdefocusing Kerr medium must be used as a host [8]. It was also noted that if this gravity-like self-interaction is strong enough, a spatial soliton may collapse into a black hole analogue.
When the nonlinear optical effects become important, they are described in terms of various order nonlinear susceptibilities  n of the metamaterial Taking into account these nonlinear terms, the dielectric tensor of the metamaterial (which defines its effective metric) may be written as ...
Similar to general relativity, Eq. (4) provides coupling between photons and the effective metric of the metamaterial "spacetime". In a central symmetric material the second order susceptibility equals zero, while the third order terms may provide correct coupling between the effective metric and the energy-momentum tensor in the weak 6 field limit. These terms are associated with the optical Kerr effect [8]. Indeed, in the weak gravitational field limit the Einstein equation is reduced to 00 4 00 2 00 8 2 where is the gravitational potential [16]. Since in the effective optical spacetime the timelike coordinate is identified with the optical axis direction z, and g 00 is identified with - 1 , comparison of Eqs. (3) and (6) performed in [8] indicated that the effective gravitational constant * may be identified as (see Eq. (16) from [8]). Since  zz = 2 < 0, in the nanowire array metamaterial, the sign of  3 must be negative for the effective gravity to be attractive. Extraordinary light rays in such a medium will behave as 2+1 dimensional world lines of gravitating (and therefore self-gravitating) bodies, and these rays may collapse into sub-wavelength spatial solitons, which are somewhat analogous to black holes [8]. Because of the large and negative thermo-optic coefficient inherent to most liquids, heating produced by partial absorption of the propagating laser beam translates into a significant decrease of the refractive index at higher light intensity. For example, reported thermo-optics coefficient of water reaches n/T= -5.7x10 -4 K -1 [17]. Therefore, ferrofluid-based selfassembled hyperbolic metamaterials appear to be an ideal material choice to demonstrate these effects in the experiment. The beam filamentation of the extraordinary light may be revealed more clearly in the differential images, in which the average Gaussian profile of the laser beam is subtracted from the currently observed beam shape. An example of such a differential image is presented in Fig. 3a. It reveals just the filaments, which are present in the extraordinary beam at a given moment of time. It is interesting to note that FFT analysis of Fig. 3a, which is shown in Fig. 3b, reveals somewhat perturbed hexagonal symmetry in the spatial distribution of the filaments. Such a hexagonal symmetry in filament distribution is quite common in other nonlinear optical systems exhibiting self-focusing and soliton array behaviour [18]. The highly symmetric spatial pattern of filaments is better revealed after high pass filtering of Fig. 3a, which is performed in Fig. 3c.
As illustrated in Fig. 4, these results demonstrate emergence of the "gravitational arrow of time" in the self-assembled hyperbolic metamaterial, which is directed along the time-like spatial z coordinate (aligned with the optical axis of the metamaterial).
Indeed, as demonstrated in [9], in the presence of gravity a simple Newtonian system of 8 dust particles develops progressively more complex structures, so that "the growth-ofcomplexity arrow" always points away from the unique past. It is very natural to identify the arrow of time with the direction in which the complexity of the structures grows. As emphasized in [9], complexity is a prerequisite for storage of information in local subsystems and therefore the formation of records. Under the action of gravity typical systems break up into disjoint subsystems which get more and more isolated, and one can associate dynamically generated local information with them. This time evolution is depicted in Fig. 4a for an example of a planetary system forming from a structureless dust cloud. The observed changes in shape of the extraordinary light beam as a function of z coordinate (depicted in Fig. 4b) exhibit virtually similar behaviour.
The original structureless Gaussian beam incident onto the cuvette separates into multiple filaments under the action of effective gravity acting in the effective 2+1 dimensional optical spacetime, in which the role of time is played by the spatial z coordinate.
On the other hand, as evident from the Supplemental Video 2, the spatial pattern of filaments exhibits fast variations as a function of the conventional physical time on the millisecond time scale. For example, four image frames taken from such a video (see Fig. 5) show gradual motion of two filaments towards each other, followed by merging of these filaments. In addition, absorption of the CO 2 laser light in the metamaterial causes gradual heating of the system, thus leading to irreversible evolution of the system as a function of the conventional physical time. Fig. 6 shows an example of such an entropic irreversible evolution of the extraordinary beam following the opening of the CO 2 laser shutter. The average temperature of the field of view measured by the LWIR camera increases from frame to frame. Note that mutual attraction of individual filaments may also be seen in this set of images. 9 Our experimental observations clearly reveal the 2T character of the ferrofluidbased hyperbolic metamaterial system. As theoretically predicted in [1], propagation of extraordinary light in this system is described by two time-like coordinates. In addition to the conventional physical time, the spatial direction aligned with the optical axis z of the metamaterial also has a time-like character, which is revealed by the Minkowskilike 2+1 dimensional metric of the optical space described by Eqs. demonstrated that the issues related to causality in such multi-time models may be resolved by compactification of the additional temporal dimensions. We note that the 2T physics of ferrofluid-based hyperbolic metamaterials considered above is consistent with this picture because of high losses, and therefore very short propagation length of the extraordinary photons inside the metamaterial. We also note that practical applications of the 2T systems may be also quite interesting. For example, they may be applied in novel optical hyper-computing schemes [20], which map a computation 10 performed during a given period of time onto a much faster computation performed using a given spatial volume of a hyperbolic metamaterial. Such hyper-computing schemes may be useful in time-sensitive applications.       Supplementary Video 1. Experimentally measured temporal dependence of the CO 2 laser beam shape for the ordinary light passing through the ferrofluid subjected to external DC magnetic field. This video was recorded at 160 mW incident CO 2 laser power.

Supplementary Video 2.
Experimentally measured temporal dependence of the CO 2 laser beam shape for the extraordinary light passing through the ferrofluid subjected to external DC magnetic field. This video was recorded at 160 mW incident CO 2 laser power.

Supplementary Video 3.
Experimentally measured temporal dependence of the CO 2 laser beam profile for the extraordinary light passing through the ferrofluid subjected to external DC magnetic field. This video was recorded at 160 mW incident CO 2 laser power.
14 Supplementary Video 4. Experimentally measured temporal dependence of the CO 2 laser beam profile for the extraordinary light polarization recorded at 22 mW incident CO 2 laser power. At this lower power the beam filamentation virtually disappears, so that the beam shape may be characterized as slightly perturbed Gaussian profile.