A Turing machine with wave memory

Computers possess a number discrete memories containing digital information. This quantization is fundamentally related to the fact that the information is stored using discrete entities, such as electrons for computers or DNA bases for living organisms. Is it possible to imagine a memory storing an analog information even at the elementary level? This is what we have shown with “walkers”. These are unique objects composed of a self-propelled droplet bouncing on a vertically vibrated liquid bath, driven by the waves its emits.

The droplet propagates on the bath surface piloted by the wave field created by its previous bounces. At each impact, the droplet bounces on a slanted liquid surface resulting from the presence of the wave packet. Thus, each bounce on a local slope give the droplet small horizontal kick which iteratively define its trajectory. It is important to note that the wave field itself results from the position of previous bounces of the drop and therefore from the trajectory itself. Each bounce creates a circular wave centered on the point of impact. However, unlike a pebble thrown in a pond, the emitted wave remains stationary around the impact point, sustained for a given time. This is a consequence of the vertical oscillations of the bath and the vicinity of the Faraday instability threshold. This parametric instability produces stationary waves at half the excitation frequency. Here, the propagating wave produced at each bounce by the droplet triggers locally this instability, creating a circular standing wave centered on the impact point. Since the bath excitation remains below the Faraday instability threshold, these standing waves are sustained only of a finite and tunable time which depends on the distance to the Faraday threshold. The global wave field is thus the result of the interference of these elementary circular waves deposited along the droplet trajectory. Thus the wave field holds the signature of the droplet trajectory. We have called this wave driven dynamics as a path-memory dynamics.

Walkers behavior can be analyzed in terms of information processing. They can perform the basic operations of writing, reading and erasing. These 3 basic operations are required to perform an elementary « Turing machine », the basis of all mechanical computing devices, such as a computer with memory. At each bounce, the drop writes in the wave field one elementary bit of information in the form of a circular stationary wave centered on the point of impact. This information is stored during a controllable tunable memory time. The droplet reads the information stored in the wave field as its dynamics results from the wave field.

This series of experiments conducted by Stéphane Perrard showed that the erasing operation could also be added to the walker dynamics to obtain a complete Turing machine. A small additional vertical jolt on the bath could induce a  shift between the droplet bounces and its wave field. The surface slope is changed into its opposite, and the droplet backtracks its trajectory erasing the information deposited previously in the wave field.

The fact that the elementary information is a wave, i.e. continuous, while associated to a single object, i.e. the walker, is very intriguing out of the quantum world. In classical physics, discrete objects usually are not endowed with wave characteristics. Here, the walkers possess a classical wave-particle duality which results from the intertwined dynamical interaction between the droplet and its wave field. Note that, although this duality is purely « classic », several previous studies showed that walkers displayed several features that were thaught to be part of the quantum world only.

The analysis of the walker’s dynamics in terms of information exchanges highlight the fact that this object possesses an internal wave memory. Thus, it can be considered as non-local in space and in time. The possibility for the walker to revisit its past trajectory, even when chaotic, for a characteristic time associated to its memory time raises fundamental issues about time reversal. While it is known that one can make the waves revisit their past life even in complex environments, it is much more difficult when it comes to particles because of the high sensitivity to initial conditions. Thus, a small error in the inversion of the velocity of a particle leads to rapidly diverging return paths. For walkers, their wave driven dynamics show a new behavior with the ability to be time reversed for characteristic times related to their temporal extension.

Finally, it is also interesting to look at these results in the light of the latest developments in computer science and the emergence of a new paradigm of « machine-learning » based on the use of complex networks to perform computation. These methods referred to as « reservoir computing », inspired by the brain’s ability to process information, exhibit state-of-the-art performances for processing empirical data such as speech recognition. These techniques are based on the use of complex dynamical systems with delayed feedback imitating the functioning of neuronal networks. Since space-time non-locality in processing of information is central both in these methods and in the walker’s dynamics, maybe this could open up new horizons.

Space-time transformations

Time reversal and holography with spacetime transformations in water waves.

Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the effective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves introducing a universal framework which explains the effect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these « Cauchy sources » which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control. This work is the subject of Vincent Bacot’s thesis.

Holographic methods are based on the time-reversal invariance of wave equations. They rely on the fact that any wave field can be completely determined within a volume by knowing the field (and its normal derivative) on any enclosing surface. Hence, information reaching the 2D surface is sufficient to recover all information inside the whole volume. Based on these properties, Denis Gabor introduced the Holographic method, which provides an elegant way to back-propagate a monochromatic wave field and obtain 3D images. More recently, time-reversal mirrors exploited the same principles extended to a broadband spectrum to create time-reversed waves. This latter approach has been implemented with acoustic, elastic, electromagnetic and water waves. It requires the use of emitter-receptor antennas positioned on an arbitrary enclosing surface. The wave is recorded, digitized, stored, time-reversed and rebroadcasted by the antenna array. If the array intercepts the entire forward wave with a good spatial sampling, it generates a perfect backward-propagating copy.

Here, within the general concept of spacetime transformations, we completely revisit the holographic method and introduce a new way to create wideband time-reversed wave fields in 2D or 3D by manipulating time boundaries. It is the dual time equivalent of standard time reversal based on spatial boundaries. We use a sudden modification of the wave propagation properties of the medium to create a time-reversed wave. This time disruption realizes an instantaneous time mirror (ITM) in the entire space without the use of any antenna or memory. The information stored in the whole medium at one instant plays the role of a bank of memories.


In this study, we use gravity-capillary waves to implement the concept of ITM. Because the surface wave celerity depends on the effective gravity, the disruption of the celerity is achieved by applying a vertical jolt to the whole liquid bath. A bath of water is placed on a shaker to control its vertical motion. A plastic tip fixed on another shaker is used to hit the liquid surface and generate a point source of waves at time t=0. The movie shows a typical time sequence of the vertical tip and bath motions used to generate the surface waves and implement the ITM from above and from the side of the bath. A circular wave packet centered on the impact point is emitted as the tip hits the surface. After time 60 ms, a vertical downward jolt is applied to the bath. The bath acceleration reaches approximately -20g in approximately 2 ms. The propagation of the initial outward-propagating wave is not significantly affected by this disruption. However, at the time of the disruption, we observe the apparition of a backward-converging circular wave packet that diverges again upon trespassing the original impact point source.

The above movie shows the evolution of the profile of a wave packet. The wavelength spreading induced by dispersion is clearly visible. The ITM generates a time-reversed wave packet propagating in the opposite direction. The resulting surface profile can be decomposed into the two counter-propagating wave packets. In contrast with standard reflection, the backward wave packet is not spatially reversed. The time-reversed nature of the backward wave allows the wave packet to compensate for dispersion. The fast short wavelengths will catch up with the slow long wavelengths, thus refocusing the wave packet. ITM is a broadband time reversal mirror.



The above movies show two examples of ITM performed on sources with a complex shape of an Eiffel Tower and a Smiley. The ITM disruption occurs long after the wave field has lost any resemblance to its initial shape at the time of emission. The wave refocus back to its initial shape indicates the time reversal nature of the process.

This work is funded by the AXA Research Fund.

Supercritical Angle Fluorescence Microscopy

Principle of SAF microscopy

Principle of SAF microscopy

Fluorophores emission directions dependent on their distance to the surface of the glass slide. The fluorophores are dipolar nano-antennas. They possess an evanescent electromagnetic field (near-field) which does not radiate in an homogeneous environment.  However, when an interface is present within this near field (i.e. a few tens of nanometers), a portion of the near-field becomes propagative. For instance, at the interface between water and glass, this emission is transmitted into the glass in directions beyond the critical angle. Thus, while the subcritical emission is identical whatever the fluorophore / glass distance, only the fluorophores located in the immediate vicinity of the interface have such supercritical emission. This light sometimes referred to as « forbidden light » (because it does not satisfy the Snell-Descartes law) may represent up to 50% of the light transmission into the glass. This supercritical emission decreases very rapidly with the distance to the interface (approximately exponential) and thus provides an absolute location of nanometric axial fluorophores. Current commercial microscope objective with high numerical aperture can collect most of this supercritical light.

We have developped new fluorescence microscopy techniques which takes advantage of this supercritical angle fluorescence (SAF) to acquire an absolute axial localization with a precision of a few nanometers.

This work is developped in collaboration with Sandrine Lévêque-Fort’s group at the Institute of Molecular Sciences of Orsay (ISMO), France.


SAF Microscopy is an ideal technique for conventional wide field microscopy as it provides a direct access to the real-time observation of membranes and adhesion processes. It offers many advantages compared to technical currently offered commercially, particularly Total Internal Reflection Fluorescence (TIRF) microscopy. It is based on a complementary approach, the spatial selectivity of not contained by a confined excitation but by a selectivity at the collection. This has many advantages including: enhanced confinement and significant background reduction avoiding intrinsic light scattering in biological media, homogeneous and straighforward illumination (without the need of laser), simultaneous real-time acquisition of the standard epi-fluorescence. This technique can also be coupled with most existing microscopy techniques. The development of SAF detection with full-field fluorescence microscopy is the subject of Thomas Barroca thesis.

A super-resolved 3D image example is shown below. Actin of the cell is labeled with the Alexa 647, on the left representing the image diffraction limited subnetwork is not discernable in contrast to the super-resolved image (right part of the image)

A super-resolved 3D image example is shown below. Actin of the cell is labeled with the Alexa 647, on the left representing the image diffraction limited subnetwork is not discernable in contrast to the super-resolved image (right part of the image)

This SAF based technique is compatible with super-resolution microscopies. In particular Single Molecule Localization Microscopies (SMLM). The association of SAF detection with SMLM is the subject of Nicolas Bourg thesis (ISMO). This new nanoscope called DONALD for Direct Optical Nanoscopy with Axially Localized Detection achieves an isotropic localization precision below 20 nm (see the ArticleDonald for details)


This work is funded by the French National Research Agency and the DIM C’Nano of the Region Ile de France and AXA Research Fund.