General background

Quantum Chaos

General introduction

Classical chaos implies exponential sensitivity to initial conditions observed in almost all nonlinear classical systems with more than 1-degree of freedom, when the number of constants of motion are less than the number of degrees of freedom, nonintegrable systems. Here is a movie of about 40 triple pendula that almost start with the same initial condition only for it to segregate radically after a short Lyapunov time. This is one important source of unpredictability and the origin of chance in a deterministic Newtonian framework of physics.


Quantum mechanics is on the other hand a very different kind of theory where the linear Schrodinger equation, a partial differential equation holds a central place. It is deterministic without measurements, but on the other hand noncommutativity of position and momentum implies that the phase space does not exist except as a pseudo phase space where coherent states of minimum finite size of Planck constant play the role of phase space "points". Interference effects wash out the infinitely variegated and structured phase space structure that is seen in classical systems and what remains is a random wave like phenomena. Quantum chaos is the study of nonintegrable quantum systems and the interest is of course that the phenomenology is so different from the classical and provides also a window into fundamental aspects of quantum-classical correspondence. Shown in two sample movies is the contrast between classical evolution of an ensemble of initial conditions and the corresponding quantum coherent state under the dynamics of a chaotic kicked pendulum.

For more information and details I link my notes last updated for an

ICTS school on Statistical Physics in the Summer of 2018.

Files for a colloquium on Quantum chaos and Quantum Information given at IIT Madras are available for download from here. Almost all the illustrations and animations are made by me using codes in the notes, use them at your discretion (and risk!). I have provided the mp4 files separately as on they are not integrated into the pdf file.

For a nice introduction to the field see Professor Steven Tomsovic's group website at WSU Pullman.

A number of research directions have been taken over the many years, I highlight a few aspects.

OTOC

First, the recent works on a topic that is called Out-of-time-ordered correlators or OTOC that seeks to find quantum equivalents of Lyapunov exponents and have been evaluated for unlikely objects such as a quantum black hole and for some quantum field theories. OTOC's are most simply thought of as growth of non-commutativity among two unequal time operators.

Our work uses almost exactly solvable quantum chaotic models to evaluate analytically the OTOC and also study it for few particle large N models to explicate the role of multiple Lyapunov exponents and the role of interactions in determining post-Lyapunov growth rates:

Solvable few body quantum chaos

A canonical example of quantum chaos, the "kicked top" was solved for "few-qubit" versions corresponding to total angular momentum j=3/2 and j=2. Surprisingly many aspects of quantum chaos and its connections to entanglement and thermalization is already present in an embryonic manner in these systems. An experiment with 3 qubits superconducting Josephson junctions ("transmons") realized the j=3/2 case and we have also analyzed their experimental data. Although they declare this model as non-integrable we showed not only that it can be construed as a special case of an integrable model, but went on and explicitly solved it. It must be noted though that chaos has nothing to do perse with solvability and that the issue of integrability of finite dimensional systems is a prickly one.

Multifractal states of quantum chaos, Thue-Morse sequence and all that

Even though a bit dated, I am still fascinated by the tantalizingly close analytical solutions to a quantum chaotic system that is given by the fascinating and ubiquitous binary sequence called the Thue-Morse sequence and its fourier transform. For they also describe in detail multifractal states, how deterministic chaos can arrange for complex states but still allow for ones "scarred" by low-period periodic orbits.

Quantum information

Quantum entanglement was highlighted by Schrodinger in his reply to Einstein Podolsky Rosen's 1935 paper and is the unique quantum property that makes it appear nonclassical or strange to us. Since the 1990's it has come to be realized as a resource in major quantum information tasks such as teleportation. Its role in quantum computation remains ambiguous but there are alternatives such as cluster state computing that explicitly consume entanglement enroute to performing a computation.

Quantum chaos and Quantum information

As chaos is quintessentially classical and entanglement quantum, it seemed somewhat unlikely that the two should pay attention to each other. However, some of our early works such as:

showed that when the whole system is pure, chaos entangled subsystems efficiently and nearly maximally. Subsequent experiments such as the one with cold atoms by Choudhury et. al. in 2009 and the more recent one with with 3 qubits superconducting Josephson (2016) have shown the interplay of the "butterfly effect" with the quantum.

More recent works of ours in this direction are:


Entangling Power

While entanglement of quantum states is usually studied, the entanglement present in operators, especially time propagators which become quantum gates in circuits, is also important and interesting. It is also connected to states via the so-called Choi-Jamialkowski isomorphism and the existence of maximally entangling operators are related to existence of Absolutely-Maximally-Entangled states of many-particles. Some of works in this direction:

Many-body systems

A consequence of appreciating quantum entanglement is the ongoing efforts to understand that there is not just one type of it in a many-particle system. In particular due to monogamy, a unique quantum feature that correlations cannot be arbitrarily shared among several parties, there is great variance in the way entanglement is shared among small subsystems and those shared by macroscopically large ones. We study these issues again from the point of view of dynamics and in systems of interest to contemporary condensed matter physics, focusing both on Floquet or periodically driven systems and on conservative systems. Some of our key findings relate to ballistic growth of entanglement in special integrable systems, the multipartite nature of entanglement in nonintegrable chaotic systems, the nature of local entanglement in many-body-localization, and the use of entangling power in many-body systems.

Some of our early and recent works in this category:


Random matrix theory

Random matrix theory developed in the 1930's by Wishart for statistical applications and in the 1950's by Wigner and later Dyson, Mehta and others for applications in nuclear physics is now a vigorously growing area at the interface of mathematics and physics with bewildering range of applications including economics, almost all areas of physics, Riemann zeta function, weather forecasting, communications and signal processing ....

We use it mainly in two contexts (i) classical chaos or quantum nonintegrability implies RMT statistics. First formulated as a conjecture by Bohigas, Giannoni and Schmit in the 1980's, it has found wider applicability even in many-body contexts wherein classical limits do not apply and (ii) defining ensembles of states or operators in Hilbert spaces. A lot of work does depend or use some ideas of RMT. It is the statistical physics of quantum nonintegrability and sometimes hides behind apparent "proofs" of the foundations of statistical physics.

Our works related especially to developments of RMT may be:

Extreme value statistics in classical and quantum chaos

We also found the surprising and even now only partially understood phenomenon that multiplying random real matrices with no special symmetry (the real Ginibre ensemble) led to increasing probability that the number of real eigenvalues increased.

We have defined new ensembles of random matrices of relevance to special sets of state and quantum information in these papers:

Recent work has also focused on new gap distributions in spectra:



triple-pendulum-chaos.mp4
StdMapEnsembleK=9q0p05.mp4
StdMapCoherentTimeEvolK=9N=100q0p05.mp4