Arjendu K. Pattanayak: Summary of published research

Mapping the quantum-classical transition for nonlinear systems (including transport properties)
(2006 -- , Carleton)

• Quantum entropy dynamics for chaotic systems beyond the classical limit(PRE, 2007): Further work on the scaling hypothesis and entropy production rate (see paper with Sundaram and Greenbaum, as well as solo PRL from '99 below): The entropy production rate for an open quantum system with a classically chaotic limit has been previously argued to be independent of [h-bar] and D, the parameter denoting coupling to the environment, and to be equal to the sum of generalized Lyapunov exponents, with these results applying in the near-classical regime. We present results for a specific system going well beyond earlier work, considering how these dynamics are altered for the Duffing problem by changing [h-bar],D and show that the entropy dynamics have a transition from classical to quantum behavior that scales, at least for a finite time, as a function of [h-bar]2/D.
• Bifurcations and sudden current change in ensembles of classically chaotic ratchets(PRE, 2007): Trying to understand transport properties of nonlinear (specifically chaotic) systems. There was an interesting conjecture from Mateos [Phys. Rev. Lett. 84, 258 (2000)] that current reversal in a classical deterministic ratchet should be understood as stemming from bifurcations from chaotic to periodic regimes. This is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential. Barbi and Salerno [Phys. Rev. E 62, 1988 (2000)] disagreed: they argued that, contrary to Mateos' claim, current reversals can occur also in the absence of bifurcations. We were trying to understand ratchets in general, and looked into it a little deeper. Turned out that Barbi and Salerno's studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system. The behavior of ensembles can be quite different, depending upon their characteristics, which leaves their results open to question. In this paper we present results from studies showing how the current depends on the details of the ensemble used to generate it, as well as conditions for convergent behavior (that is, independent of the details of the ensemble). We are then able to present the converged current as a function of parameters, in the same system as Mateos as well as Barbi and Salerno. We show evidence for current reversal without bifurcation, as well as bifurcation without current reversal. We conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation, rather than current reversals, and show numerical evidence for our claims.

• Coarse-grained entropy decrease and phase-space focusing in Hamiltonian dynamics (PRA, 2005, also Virt. J. Ultrafast Sci. -- Aug 2005): We analyze the behavior of the coarse-grained entropy for classical probabilities in nonlinear Hamiltonians. We focus on the result that if the trajectory dynamics are integrable, the probability ensemble shows transient increases in the coherence, corresponding to an increase in localization of the ensemble and hence the phase-space density of the ensemble. We discuss the connection of these dynamics to the problem of cooling in atomic ensembles. We show how these dynamics can be understood in terms of the behavior of individual trajectories, allowing us to manipulate ensembles to create "cold" dense final ensembles. We illustrate these results with an analysis of the behavior of particular nonlinear integrable systems, including discussions of the spin-echo effect and the seeming violation of Liouville's theorem.
• Pulse-induced focusing of Rydberg wavepackets (PRA, 2003): We demonstrate that strong transient phase-space localization can be achieved by the application of a single impulsive "kick" in the form of a short (600 ps) unidirectional electric-field pulse to a strongly polarized, quasi-one-dimensional Rydberg atom. The underlying classical dynamics is analyzed and it is shown that phase-space localization results from a focusing effect analogous to rainbow scattering. Moreover, it is shown that the essential features of the classical analysis remain valid in a quantum-mechanical treatment of the system in terms of its phase-space Husimi distribution. The degree of phase-space localization is characterized by the coarse-grained Renyi entropy. Transient phase-space localization is demonstrated experimentally using extreme redshifted m = 0 potassium Stark states in the n = 351 manifold and a short probe pulse. The experimental data are in good agreement with theoretical predictions. The localized state provides an excellent starting point for further control and manipulation of the electron wave packet.
• Parameter scaling in the decoherent quantum-classical transition for chaotic systems (PRL, 2003): The quantum to classical transition for a system depends on many parameters, including a scale length for its action, [h-bar], a measure of its coupling to the environment, D, and, for chaotic systems, the classical Lyapunov exponent, lambda. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of ([h-bar],lambda,D) and searching for transformations that collapse this hypersurface into a function of a composite parameter zeta= [h-bar]^alpha lambda^beta D^gamma. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition. (With Sundaram and Greenbaum)
• Non-hermiticity in a kicked model: Decoherence and the semiclassical limit (PRE, Rapid, 2002): People have been studying dynamical localization in quantum chaotic systems (related to Anderson localization) for many years. It seems likely that as the decoherent effect of the environment is turned up, the localization should give way to classical behavior. We tried to understand this using the novel method of non-Hermitian Hamiltonians in a quantum kicked model exhibiting a localization transition. We showed that the critical line separating the extended and localized phases approaches its semiclassical limit as the non-Hermiticity (corresponding to the decoherence) is steadily increased. This direct evidence of quantum-classical correspondence means that decoherence may be usefully modeled by non-Hermitian perturbations. (Work done with Indu Satija).

Dynamics of probability distributions, with application to experiments
(1998-2001, Rice and Carleton)

• Transient phase-space localization (PRA, Rapid, 2002): We consider the dynamics of a Rydberg atom subject to an impulsive momentum transfer or `kick'. Classical simulations and analysis of the entropy dynamics predict that the wavepacket generated by the kick undergoes strong transient phase space localization, forming an excellent starting point for further control and manipulation. Such localized states can be `trapped' for extended periods using a train of subsequent kicks.
• Stabilizing an attractive Bose-Einstein condensate by driving a surface collective mode (PRA, 2001) : Bose-Einstein condensates with attractive interatomic interactions implode unless the number of condensate atoms is less than a maximum value. We theoretically demonstrate that the nonlinear dynamics of the condensate stabilizes such a condensate against the collapse.
• Characterizing the metastable balance between chaos and diffusion (Physica D, 2001): New diagnostics for the balance between chaos and noise for the chaotic-advective dynamics of a field in a fluid dynamical system were examined in detail. It was shown in particular that the root-mean-square Fourier radius of the field distinguishes clearly between the chaotic (increasing Fourier radius) and diffusive (decreasing Fourier radius) regimes and exhibits steady-state behavior when the two are in balance. (This paper was designated one of the 'hottest papers' at Physica D).
• Lyapunov exponents, entropy production, and decoherence (PRL, 1999) : It was proven that the entropy production rate of a classically chaotic Hamiltonian system coupled to the environment settles, after a transient, to a meta-stable value given by the sum of positive generalized Lyapunov exponents. A statistical steady state is generated in this process, arising from the balance between chaos and noise. This behavior also occurs in quantum systems close to the classical limit so that quantum-classical correspondence in chaotic systems is restored through the coupling to the environment. This analytically proves a corrected and generalized version of a conjecture by Wojciech Zurek and Juan Pablo Paz.

Chaos and Correspondence in Quantum and Classical Distributions; Decoherence:
(With Paul Brumer, at U of Toronto)

• Exponential Divergence and Long Time Relaxation in Chaotic Quantum Dynamics ( PRL 1996): The dynamics of quantal quasi-probabilities (the Wigner function) and classical Liouville probability densities were compared for maps on the torus, including the Arnold Cat Map, to show the approach to classical chaos in quantum systems. 
• Chaos and Lyapunov exponents in classical and quantal distribution dynamics (PRE 1997): An analytic and asymptotically valid signature for chaos in distributions was derived and the role of a generalized Lyapunov exponent in chaotically evolving classical distributions was established; when applied to a quantum map, it was shown that the hbar --> 0 limit yields operational quantum chaos but the quantum--classical transition is not necessarily monotonic in Planck's constant.
• Exponentially Rapid Decoherence in Chaotic Quantum Systems (PRL 1997): Decoherence is the transition of a quantum system from a `pure state' exhibiting quantum interference to a statistical mixture of states. It was shown using the above result that, in the semiclassical limit, a quantum system whose classical counterpart is chaotic decoheres exponentially rapidly. This arguably contributes to the puzzling experimental observation that the decoherence time in mesoscopic devices saturates at very low temperatures.

Gaussian Approximations; Semiclassical Quantization of Chaotic systems:
(With Bill Schieve, UT-Austin)

• Semiquantal dynamics of fluctuations: Ostensible quantum chaos (PRL 1994): A finite-dimensional variational approximation to quantum dynamics was derived from Ehrenfest's theorem using Gaussian wavepackets; this can be recast as a classical-like Hamiltonian system. An example of ostensible quantum-induced chaos within this approximation was demonstrated. 
• Gaussian wave-packet dynamics: Semiquantal and semiclassical phase-space formalism (PRE 1994): Other interesting properties, including a propagator satisfying unitarity and group behavior and the emergence of the Bohr-Sommerfeld and Maslov phases as the dynamical and geometrical versions respectively of Berry's phase, were developed. These dynamical equations were compared to the standard Gaussian semiclassical dynamics; an extended and improved semiclassical dynamics was introduced. 
• Calculation of eigenvalues of a strongly chaotic system using Gaussian wave-packet dynamics (PRE 1997): An accurate spectrum for a classically strongly chaotic Hamiltonian was obtained within this formalism and broad arguments for the validity of the approximation considered. 
• Effective potentials and chaos in quantum systems (PRA 1992): A further adiabatic approximation to the variational equations yields the Gaussian Effective Potential (GEP) which shows its dynamical constraints. The GEP was used in the Henon-Heiles problem to show the quantum suppression of chaos. 
• Predicting 2-dimensional Hamiltonian chaos: (Zf A, 1996)A criterion for onset of chaos in classical Hamiltonian systems based on the Jacobi equation for the separation of geodesics was proposed and successfully applied. 
• Semiquantal corrections to stochastic resonance (PRE 1995): It has been calculated using the GEP that quantum corrections to stochastic resonance shift the peak and enhance the amplitude of the signal-to-noise ratio versus noise strength curve.
• Gaussian effective potential: Computation of eigenvalues (PRA 1996): Semiclassical eigenvalues using the GEP have been computed in some simple cases; these are show to improve upon the results of the first-order Bargmann-Voros semiclassical method to which the GEP is intimately connected.
• Semiquantal and semiclassical dynamics and chaos A conference report on thesis work.

Complex Systems:
(Work done at the Santa Fe Institute and UT-Austin)

As a student at a Santa Fe Institute Summer School, I was exposed to a variety of ideas in `complex systems' including topics in neural networks, biological neurons, self-organized criticality and fluid dynamics. I worked on the following problems:

• Cellular Automata with Changing Radii: A Model for Intraspecific Computation in Plant Populations: Cellular automata with changing radii were introduced to understand the effects of regular versus random initial conditions in a model of intraspecific competition in plant populations.
With Alfred Hubler, UIUC

• Blood Coagulation is a Complex System: A cell-based experiment to understand the mechanism of blood coagulationin vivo was modelled by a system of kinetic equations to be used for numerical comparison with the experimental results.
With Maureanne Hoffman (Duke Medical School) and others. 

• Symmetry breaking through autocatalysis has been shown using a simple chemical model. With David Lippmann (Southwest Texas State and UT-Austin).

While in Austin:

• Bill Schieve (Ph.D. advisor),
• Doug Reale (Bill's Masters student),
• David Lippmann (met as an associate of the Stat Mech Center),
• Alfred H\"ubler (met at the Santa Fe Institute),
• Maureen Hoffmann,
• Bill Fortin,
• Doug Monroe (met all three at the Santa Fe Institute)

While in Toronto:

• Paul Brumer (post-doc advisor),
• Josh Wilkie (Fellow member of Paul's group).

• Randy Hulet (Rice Physics),
• Cass Sackett (Member of Randy's group),
• Arnaldo Gammal (Visiting Randy's group)
• Barry Dunning (Rice Physics),
• Chris Stokely (Member of Barry's group),
• Carlos Reinhold(ORNL, Barry's collaborator)

• Indu Satija (Met at a conference)
• Barry, Chris, Carlos,
• Diego Arbo (Carlos's post-doc) again,
• Joachim Burgdorfer (Carlos and Barry's collaborator),
• W. Zhao (Member of Barry's group),
• Jim Lancaster (Member of Barry's group)
• Bala Sundaram (Known for years through conferences/short visits),
• Ben Greenbaum (Bala's student from Columbia) and
• Anton de la Fuente,
• Dan Krawisz,
• Ted Holby,
• Jorge Silva,
• Lawrence Uricchio,
• Dan Brooks (all research students),
• Arnaldo again,
• Anatole Kenfack (met during sabbatical visit to MPIPKS),
• Sean Sweetnam (Research student)

At Rice:

• David Pekker --> UIUC (Solid State, Goldbart's group)
• Roy Keyes --> UNM
• Will Ray --> UMD (Nonlinear Dynamics/Lasers -- Raj Roy's group)
• Austin Cottrell --> UT Austin TICAM

• Anton de la Fuente ('03)
• Daniel Krawisz ('04)--> UT Austin,
• Ted Holby ('04)--> UWisc Materials,
• Jorge Silva ('04)--> Manchester,
• Lawrence Uricchio ('05) --> Chicago Biophysics,
• Dan Brooks ('05)--> UC Berkeley (Cold atoms -- Dan Stamper-Kurn's group),
• Charlotte Christensen ('05)--> UWashington Astro,
• Neal Meyer ('06) --> Penn State,
• Mark Knight ('06) --> Rice (Nanotechnology -- Naomi Halas' group),
• Leigh Norris ('07) --> U New Mexico (Quantum Information),
• Sean Sweetnam ('08),
• Parin Sripakdeevong ('08),
• Adam Steege ('08)
• Chris Amey ('09).

Long-term visits, including sabbatical visits