Arjendu K. Pattanayak: Research

Our universe is fundamentally quantum mechanical, yet truly quantum behavior isn't easily observed. You have to go into a lab and set it up right, to see the `quantumness' of a system. Or else it's transient (that is, lasts for a short time). The way we currently understand this is that the time-scales for quantumness to go away are set by how isolated you can make the system you are looking at from all other systems, i.e. `the rest of the Universe'. Along with a growing understanding of quantum systems and our ability to manipulate them, we need to characterize and control how this `isolation timescale interacts with the internal time-scales for the system'. That is, these are fundamentally non-equilibrium quantum statistical systems, and there are specific applications in atomic physics, quantum optics, and condensed matter.

Such systems are particularly exciting when the quantum system displays unusual nonlinear behavior. For example, experiments involving cold quantum gases in nonlinear potentials have been exploring fascinating new phenomena such as time-dependent chaos-assisted tunneling and dynamical localization. As the nonlinear effects of quantum many-body effects are added to the mix, this enables the exploration of even more exotic phenomena. Similarly, devices such as nonlinear nanomechanical oscillators are becoming cleaner, entering the regime where the quantum effects are very important. The nonlinearity itself either emerges most cleanly in the near-classical regime or can be understood in terms of how the classical nonlinear dynamics affects the quantum dynamics. Mapping the connection between the classical time-scales and the quantum system is then fundamental to understanding nonlinear quantum systems and to devising methods for their control.

The bottom line is that I am exploring the quantum and classical dynamics of probabilities in nonlinear (possibly chaotic) systems. In studying these things, we have to think carefully about the impact of coarse-graining and noise (environmental perturbations). This involves understanding the interplay of three different kinds of 'probabilistic' behavior -- stochastic or random, chaotic, and quantal.

My research career started by comparing quantum and classical dynamics in chaotic systems, and I have residual interest in reliable approximations to quantum nonlinear dynamics (including the nonlinear Schrodinger equation).Also, I have just begun to find my way back to research in complex systems.

Tools: I am a pen and paper theorist who believes that numerical experiments on computers to enhance intuition, verify analysis and in general serve as a 'third way' of understanding nature. As problems get more complex, computational tools are by definition more important. We ( Dani Kohen and I) have recently built a parallel computing cluster of the kind known as a Beowulf -- we call ours the Wulfhorde. The Research Corporation provided me with 2/3 of the funds used for the cluster.

I was a KITP Scholar (2003-2005) at the Kavli Institute of Theoretical Physics in Santa Barbara, CA.

Summary of previous work including links to almost all published papers, and a list of co-authors.

Hunting quantum butterflies: The quantum-classical transition for chaotic systems [Online talk (2007) at the Kavli Institute for Theoretical Physics, Santa Barbara]

Chaos, quantum mechanics and noise: Some answers, more questions [Notes for a talk 11/98].

• Decoherence and parameter scaling:

  It has been shown that in general the difference between quantum and classical behavior for chaotic systems is strongly noise-influenced. I am trying to understand its dependence on system parameters. This transition must be affected by the degree of chaos in the limiting classical Hamiltonian, the appropriately scaled value of hbar (Planck's constant) for the system, the strength of environmental perturbations, and the initial conditions. As such, the parameter space should have a rich structure, and may be non-monotonic as a function of any one parameter. Some results in collaboration with Bala Sundaram of CSI-City University of New York argued, and showed for some specific simple systems that there is interesting scaling behavior in this transition -- consistent, for example, with the experiments on decoherence and chaos done in Mark Raizen's lab at UT-Austin. There some other papers now out in the literature that show that this is not a bad speculation in a few other cases, but we certainly don't understand it very well.

  I am working on this further with Arnaldo Gammal of Sao Paulo. We are looking at this using the Wigner function formalism, as well as the fascinating new formalism of `quantum trajectories'.

• Chaos in quantum state diffusion/quantum trajectories:

  There's a 'new' formalism which considers interesting ideas such as 'weak measurement' and environmental effects to yield a stochastic Schr\"odinger equation with trajectories in Hilbert space. These trajectories have been shown to be chaotic in some situations. I am working with Arie Kapulkin, who I've known since grad school, to understand the whole new world of phenomena this opens up. Mark Knight ('06) is helping us immensely with some of the computational aspects.

• Classical chaos and the behavior of the probability density:

  The linear or Renyi entropy of second order [ln(Tr(rho²))] also measures local probability density or field intensity in classical systems, leading to different applications for entropy dynamics.

  For example, states resembling the statistical steady states that I believe arise from the balance between chaos and noise have been observed in beautiful 2d-fluid dynamics experiments by Jerry Gollub and students at Haverford College. Details of their behavior are poorly understood. I have been trying to understand this better for a few years.. recent results with Bala and his colleague Drew Poje suggest that we might finally have something of a handle on it.

• Entangled chaos:

  Entanglement is a weird property of quantum systems where things that seemingly should have nothing to do with each other get correlated. It is very counter-intuitive (it's what caused Einstein to essentially reject quantum mechanics) and is also a potentially powerful resouce (it is argued to be at the heart of the power of quantum computing). There is evidence that entanglement works very differently in non-linear/chaotic systems; I am investigating this further in collaboration with Ivan Deutsch and his student Collin Trail of the Deutsch Group at University of New Mexico. Parin Sripakdeevong ('09) is working on the classical parts of this, and Leigh Norris ('07) has just come on board to look at the quantum entanglement part.

• Quantum ratchets:

  My friend Anatole Kenfack and I have recently started playing with some of the unusual transport properties of these nonlinear chaotic systems. It's early times and we're still learning how these things work. I've gotten Sean Sweetnam ('08) involved in some of the preliminary classical calculations.

Future directions /Back-burner :

Bose-Einstein condensation Bose-Einstein condensation (BEC) in cold and dilute atomic gases are an intriguing new domain for studying nonlinear quantum systems -- the recent excitement in this field was capped by the 2001 Nobel prize.

This opens up a whole new dimension of nonlinearity in quantum dynamics, and there promises to be some very interesting physics, including the possibility of chaos and macroscopic (with 1000+ atoms involved) quantum tunneling.The general issue of chaos and decoherence in Bose-condensates is part of my long-term project with Arnaldo.

One of these days, Paul Brumer and I are going to sit down and work out what we know about the relationship between microscopic dynamics and mesoscopic (statistical mechanical) time-scales.