Trapping charged particles in free space has immense applications in quantum computing,
mass spectroscopy and atomic physics. A simple device to achieve this objective was invented
by the 1950s for which W. Paul and Hans G. Dehmelt were awarded the Nobel Physics Prize in 1989.
This device is usually called the Quadrupole Ion Trap (or simply, Paul Trap) and uses spatially
linear time-periodic electric fields. There are several open questions in this area that need
to be addressed from both the theoretical and practical aspects. The dynamics of a single charged
particle in such traps is now well understood. But the moment we have two or more particles,
the situation becomes too complex with several yet unexplained observations. The most significant
of these is what is known as RF heating, wherein charged particles in such traps are found to
acquire a significantly higher temperature than the background gas. Our primary interest in this
area is in the modelling of charged particle dynamics in such traps with the aim of someday being
able to explain RF heating. Here are two of our selected publications in this area:

RF heating of plasmas is of importance not only in Paul traps, but also in capacitive discharges, which are generated when a sufficiently high RF voltage is applied across two capacitor plates. Though a detailed study of these capacitive discharges is somewhat intractable, important insights can be gained through a simple mathematical model called Fermi Acceleration, which mimics the charged particle dynamics in these systems. This model is also of immense importance in the study of dynamical systems and classical mechanics in mathematics. Fermi acceleration essentially consists of the dynamics of a particle within a dynamical billiard of arbitrary shape undergoing periodic oscillations. The particle moves in a straight line with constant velocity between two elastic collisions with the billiard boundaries. The particle's energy changes at each collision with the moving parts of the boundary and remains same during collisions with the static parts of the boundary. The primary question is : Does the particle gain energy unboundedly with time? We have shown that certain pseudo-integrable billiards with oscillating boundaries can lead to exponential-in-time growth of particle's energy. This is quite significant from the experimental perspective since anything slower than exponential can be easily annihilated by dissipation.

The problem of Fermi acceleration assumes the billiard boundaries to be infinitely massive, i.e. the energy of the oscillating boundary does not change as a result of collisions with the particle. A related and perhaps more important question is regarding equilibration of energies in such slow-fast systems when the billiard boundary also has a finite mass and is attached to a linear spring. In this case, the particle's energy does not grow unboundedly since the total energy of the system remains constant. It has been proven that, if the billiard is chaotic, the energies of the particle and the oscillating boundary will not equilibrate for a very long time. To find conditions of equilibration in such systems has been a long-standing open problem in statistical mechanics. We have recently shown in a paper published in PNAS that the energies do equilibrate if the frozen billiard is non-ergodic and non-integrable.

## Research Areas

kushals[AT]iiserb.ac.in

Trapping charged particles in free space has immense applications in quantum computing, mass spectroscopy and atomic physics. A simple device to achieve this objective was invented by the 1950s for which W. Paul and Hans G. Dehmelt were awarded the Nobel Physics Prize in 1989. This device is usually called the Quadrupole Ion Trap (or simply, Paul Trap) and uses spatially linear time-periodic electric fields. There are several open questions in this area that need to be addressed from both the theoretical and practical aspects. The dynamics of a single charged particle in such traps is now well understood. But the moment we have two or more particles, the situation becomes too complex with several yet unexplained observations. The most significant of these is what is known as RF heating, wherein charged particles in such traps are found to acquire a significantly higher temperature than the background gas. Our primary interest in this area is in the modelling of charged particle dynamics in such traps with the aim of someday being able to explain RF heating. Here are two of our selected publications in this area:

Analytic, nonlinearly exact solutions of an rf confined plasma

K. Shah and H. S. Ramachandran, Physics of Plasmas 15, 062303 (2008)

Time evolution of Tsallis distribution in Paul trap

V. Saxena and K. Shah, IEEE Trans. Plasma Science 45, 918 (2017)

RF heating of plasmas is of importance not only in Paul traps, but also in capacitive discharges, which are generated when a sufficiently high RF voltage is applied across two capacitor plates. Though a detailed study of these capacitive discharges is somewhat intractable, important insights can be gained through a simple mathematical model called

Fermi Acceleration, which mimics the charged particle dynamics in these systems. This model is also of immense importance in the study of dynamical systems and classical mechanics in mathematics. Fermi acceleration essentially consists of the dynamics of a particle within a dynamical billiard of arbitrary shape undergoing periodic oscillations. The particle moves in a straight line with constant velocity between two elastic collisions with the billiard boundaries. The particle's energy changes at each collision with the moving parts of the boundary and remains same during collisions with the static parts of the boundary. The primary question is : Does the particle gain energy unboundedly with time? We have shown that certain pseudo-integrable billiards with oscillating boundaries can lead to exponential-in-time growth of particle's energy. This is quite significant from the experimental perspective since anything slower than exponential can be easily annihilated by dissipation.Exponential energy growth in a Fermi accelerator

K. Shah, D. Turaev and V. Rom-Kedar, Physical Review E 81, 056205 (2010)

The problem of Fermi acceleration assumes the billiard boundaries to be infinitely massive, i.e. the energy of the oscillating boundary does not change as a result of collisions with the particle. A related and perhaps more important question is regarding equilibration of energies in such slow-fast systems when the billiard boundary also has a finite mass and is attached to a linear spring. In this case, the particle's energy does not grow unboundedly since the total energy of the system remains constant. It has been proven that, if the billiard is chaotic, the energies of the particle and the oscillating boundary will not equilibrate for a very long time. To find conditions of equilibration in such systems has been a long-standing open problem in statistical mechanics. We have recently shown in a paper published in

PNASthat the energies do equilibrate if the frozen billiard is non-ergodic and non-integrable.Equilibration of energy in slow-fast systems

K. Shah, D. Turaev, V. Gelfreich and V. Rom-Kedar, Proc. Natl. Acad. Sci. USA 114, E10514 (2017)