Omri Gat

Office: Levin 17
Phone: (972-2) 6586361
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Master's and Ph.D. projects are available in a fascinating new theoretical research topic involving semi-classical dynamical systems. Contact me for more details.

Research Areas:

Nonlinear dynamics and statistical physics of ultrashort laser pulses

The coherent light generated by multimode lasers can be compressed into extraordinarily short pulses when the phases of the lasing modes are aligned. These pulses, where light is comressed into a high intensity short signal, are used in a wide range of applications ranging from optical communications networks to probing basic processes such as real time atomic dynamics. The formation and  evolution of the pulses is therefore an interesting and important area of nonlinear optics and nonlinear dynamics in general.
However, the laser light is also subject to noise sources, most notably the intrinsic quantum noise generated by the optical amplifier. This noise introduces entropy into the many mode dynamical system, a disrodering agent which inhibits the formation of pulses.
Complex amplitudes of lasing modes in continuous wave (cw) operation (left) and mode locked operation (right).  The switching from cw to pulsed operation is an ordering phase transition, as in, for example, the ordering of spins in a ferromagnet.
The process of pulse formation is therefore the consequence of stochastic lightmode dynamics (SLD), i.e., many mode dynamics subject to noise. The onset of mode locking iteslf has a precise significance of a thermodynamic phase transition in a far from equilibrium setting.
An important method to achieve pulse formation is passive mode locking wherein one introduces into the the laser cavity a saturable absorber - a nonlinear passive element which becomes more transparent the higher the intensity of the incident light. Labeling the absorber saturability by  the intracavity power by P and the noise power by T,  we have shown that when the dimensionless parameter  is increased, the light mode system experiences a first order  transition from a disordered cw phase, where the invariant measure is concentrated on cw waveforms, to a mode locked phase, where probable waveforms consist of a short pulse accompanied by a weak cw background. Under certain circumstances the transition occurs when  has the precise value 9. The results are obtained using exact mean field theory.
Typical laser cavity optical waveforms in the cw phase (left) and mode locked phase (right). The fluctuations are generated by the intracavity noise, and are characterized by the coherence length. Mode locking occurs when the pulse width is shorter than the coherence length. Thus, the onset of mode locking is an abrubt discontinuous process.
The simple picture described above is valid only when the pulse peak power is not too large. When the pulse power is large eneough to saturate the absorber, the laser may reach a situation where it is more favorable to form two or more pulse per roundtrip period. Then the thermodynamics of the ligth mode system is described as a cascade of first order phase transition in a phase diagram with many phases, labeled by the number of pulses. 
Another twist is occurs when the mode locked laser is coupled to an external source of pulses. The external pulse source assumes the role of an extenral driving field in the theory of phase transitions. When the external seeding is strong enough, the cw-pulse transition disappears in a critical point. The critical point is exactly described by the Wan der Waals theory of phase transitions, and classsical exponents have been measured experimentally.

Geometric swimming and motion in classical gauge field

Swimming is a form of self propulsion, where translational or rotational motion is achieved by a series of deformations. When the swimming motion occurs in conditions of low Reynolds numbers, that is, when inertia is completely negligible compared to the viscous forces, then swimming is geometric. That is, that the total motion depends only on the series of shapes achieved by the swimmer and not on the rate of deformation. Since the Reynolds number is proportional to the size and speed of the swimmer, swimming of microscopoic objects, whether microorganisms, or microrobots is geometric. A related problem, which shares many of the propetries of geometric swimming, while being easier to analyze, is the rotation withouit angular momentum of freely suspended objects, as in the famous ability of a falling cat to land always on its feet, again achieved by a series of deformations.
Mathematically, geometric swimming is described using a connection on a fiber bundle, a concept better known in physics as a gauge field. The gauge field is defind on the space of all possible shapes - in principle an infinite-dimensional space - which can be reduced significantly by considering swimmers whose shapes are confined to a certain predefined class. For example, the swimmer shapes shown in the adjacent figure are obtained as the image of the unit circle under members of a certain two-parameter family of conformal maps. As a consequence of its up-down symmetry, the depicted two-dimensional swimmer can only move in the horizontal direction, and cannot rotate. The swimming action can be described in physical terms using a 'magnetic field' defined on the two-dimensional abstract shape space. A swimming stroke can be defined as a closed orbit, i.e. a loop, in the space of shapes, and the translation achieved by the swimmer is equal to the 'magnetic flux' through the loop. 
The research work described here focused on the question of efficiency. Namely, we asked how the swimmer can perform a certain task, such as propelling itself a given distance, using the minimum amount of energy. This question is especially pertinent for artificial swimmers, which have to work with a limited supply of energy. Since the dissipation of energy is a quadratic functional of the swimmer path, the problem of identifying the optimal swimmer is isomorphic to Hamilton's principle of least action in the clasasical mechanics of a charge in magnetic field, where the 'kinetic energy' is defined by a Riemannian metric derived from the energy dissipation functional. The shapes shown on the right are a series of snapshots from the solution of the mechanical problem in a conveniently picked space of shapes. The swimmer advances horizontally, and the snapshots are shifted vertically only for visibility. The swimming stroke they describe is the most energitically efficient one in the class of shapes studied.



Fractal spectra and magnetic oscillations in lattices threaded with a magnetic field

The Harper-Hofstadter system is one of the paradigms of solid state physics. First cosidered by Peierls, it describes non-interacting fermions hopping on a two-dimensional lattice and subject to a transverse magnetic field. In the tight-binding approximation the Hamiltonian is periodic in the magnetic field, depending only on the magnetic flux threading a unit lattice cell modulo the flux quantum h/2e The energy spectrum of the system has a very unusual dependence on the magnetic field. When the number of flux quanta is rational, Bloch's theorem applies, and the spectrum consists of a finite number of bands; when the number of quanta is irrational, on the other hand, the spectrum is a Cantor set of zero measure. When the spectrum of the Hamiltonian is plotted against the strength of the magnetic field for several rational values, the beautiful picture shown on the right emerges. It is nicknamed 'Hofstadter's butterfly'. In the figure shown, the gaps, which are of full measure because of the fractal mature of the spectrum, are colored by the associated value of the Hall conductance, which is integer and constant in each gap, because of the quantum Hall effect. 
As a consequence of the extreme sensitivity of the energy spectrum in the Hofstatdter system, physical quantities which involve differentiation with respect to magnetic field are strange and singular. This is indeed observed in the (orbital) magnetizartion, the derivative of the free energy with respedct to the magnetic field. At zero temperature the magnetiuzation in the gaps is an oscillating piecewise linear function of the Fermi energy, whose frequncy of oscillations grows without bound when approaching a magnetic Bloch band, filling a two-lobed envelope. The adjacent figure shows the magnetization oscillations as a function of the Fermi energy when the magnetic flux is close to one third. They are the lattice analog of the well-known de Haas-van Alphen magnetic oscillations predicted by Landau. The shape of the envelope reveals some subtle properties of the quantum state, such as Berry's phase, and a second quantum phase first discovered by Wilkinson, Belissard and Rammal.

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