#### Quantum classical correspondence :

Classically chaotic systems display sensitive dependence on
initial conditions. Many physical systems such as the driven
and damped pendulum, atoms in strong magnetic fields, nonlinearly
coupled oscillators and periodically driven rotors exhibit this property.
Chaos, of course, pervades anything from weather to perhaps even stock market
dynamics. See my

somewhat non-technical talk on Hamiltonian chaos for more details.

The quantum dynamics of such systems do not display dependence
on initial conditions. One recurring theme of my research is to
explore and understand the manifestations of classical chaos in
the quantum dynamics. In particular, interest is foucussed on
how the rich variety in classical dynamical structures affect the
quantum spectra. Recently, we had studied

quantum-classical correspondence
in a non-KAM system.

#### Chaos, quantum localisation and interactions :

The interplay between chaotic dynamics,
localisation and interactions is of special interest especially in the
context of driven quantum systems.
Kicked rotor, a pendulum periodically kicked by an external potential,
is a prominent and well studied chaotic system
We study different variants of kicked rotor that
throw light on the connections between the quantum dynamical localization,
a variant of the Anderson localization, and classical (sub-)diffusion. We recently wrote a detailed review
article on

*Quantum kicked rotor
and its variants: Chaos, localization and beyond * in

*Physics Reports (2022)*.

In a recent work that combines theory, numerics and cold atoms experiment (in collaboration
with my colleague

Umakant Rapol),
we have shown how specific manipulation of kick sequence imparted to a rotor can lead to

control of
quantum decoherence, which shows up as a decay of localization.

In general, the role of interactions and how they affect localization properties in
chaotic and many-body systems is explored. The physical systems of interest range from
coupled kicked rotors and kicked tops to quantum spin-chains.

#### Random matrix spectra :

Random matrix theory (RMT) is widely used in physics (in condensed matter
physics, nuclear physics etc.)
to understand the spectral signatures of complex quantum systems and
as a qualitative indicator of underlying classical dynamics. RMT is extensively
applied in the study of chaotic quantum systems. Recently,
we had obtained a

series of results
for the higher order spacing ratio distributions in random matrix theory and
complex quantum systems.

#### Quantum computing and machine learning :

Quantum computer is expected to speed-up certain class of problems,
the prime factorisation being one such example. One of the central resource for
a quantum computer is the quantum entanglement. Our research is
focussed on studying entanglement (and more generally quantum correlations) properties, and curiously, to
understand how it is affected by classical dynamics in the case
of chaotic and many-body systems. Our recent representative paper on this topic is

Phys. Rev. E **95**, 012216 (2017).
Another exciting area of interest we are starting on is the quantum machine learning (QML). Many machine
learning tasks can possibly benefit from their quantised versions. With the advent of real
quantum computers, we might actually be able to implement QML algorithms in some years.
This has both fundamental and practical benefits, as discussed in

Nature **549**, 195 (2017).

#### Complex Networks :

has emerged as a unified framework to study
many complex systems, ranging from stock markets to biological
processes. We are mainly interested in the dynamical processes on
networks, their properties and the interplay between network
structure and dynamics on networks. The dynamical processes that we study
can range from random walks to coupled chaotic systems.
These are relevant for many transport processes on networks and
also for such applications as web search engines and online recommender
systems.

#### Extreme events :

Extreme events are an interesting emergent phenomena in complex systems.
Floods, droughts, traffic jams, mobile congestion are some of the commonly
encountered extreme events. Unlike floods and droughts, the latter two
are examples of extreme events taking place on networks.
The aim is to not only understand the properties of extreme events on networks
but also explore the possibility of controllinng them. Our earlier
research in the context of random walks on networks has showed
a rather counter-intuitive property that extreme
events are more likely to take place on small degree nodes of a network
than on hubs. Our first paper on this topic is

Phys. Rev. Lett. **106**, 186701 (2011).
I co-edited a special issue on

*Extreme events and its applications* and its

introduction provides a brief overview of the subject.

#### Complex systems / Data Science :

Since the last few decades, principles of physics are
beginnning to be applied to areas outside of it. Econophysics is one such
area that represents an application of principles of physics, especially statistical
physics, to some problems of financial markets. A good review is available

here. Here, our efforts are directed towards understanding larger organising principles
of complex systems. One of our recent works was to study

record statistics of financial data
in the context of geometric random walks.
Since the pandemic began in 2020, we used extensive
mobility data in India to create an

infectious diseases hazard map.