Our primary research involves quantum (or classical wave) systems with a non-integrable classical (or ray) limit. In simple terms, we study wave behavior in complicated geometries. An example of classically chaotic dynamics could be the motion of a billiard ball inside an irregularly shaped box. The corresponding quantum (or classical wave) problem might concern the wave functions and transport of electrons in a quantum dot of that shape (or the vibration of a drumhead of that same shape, or of microwaves inside a resonator of that shape). Thus, the field of quantum chaos combines methods, insights, and examples from diverse areas, such as condensed matter and mesoscopic physics, atomic, optical, chemical, nuclear, and microwave physics, nonlinear dynamics, and mathematical physics. We aim for broadly applicable results, that will spur further theoretical and experimental progress in specific areas, while at the same time furthering our fundamental understanding of the classical-quantum interface.

Recent research directions for our group include:
Focusing and Branched Flow in Weak Random Potentials
Interaction Matrix Element Fluctuations in Quantum Dots
Structure of Wave Functions and Quantum Transport in Generic Systems
Many-Body Interacting Systems
Time Ordering and Time Correlation in Kicked Qubits
Long-Time Semiclassical Accuracy and Quantization Ambiguity
Vacuum Energy in Chaotic Geometries

Our work has been supported in part by:
Louisiana Board of Regents Support Fund (Research Competitiveness Subprogram)
National Science Foundation (CAREER award)
Tulane Research Enhancement Fund (Phase II)

Focusing and Branched Flow in Weak Random Potentials:

In a two-dimensional electron gas (2DEG), electron flow is observed to be concentrated in branches of strong current density. This is initially surprising, since the potential in which the electrons move is known to be random. Leading-order caustics or singularities in the classical flow can explain this branching, and allow statistical properties of the flow to be analytically predicted, including the growth in the number of branches and the distribution of branch heights. Outstanding questions include the effect of inter-caustic correlations, extensions to finite electron wavelength, and the inclusion of boundary effects. Currently, in joint work with Eric Heller of Harvard University we are addressing the implications of a similar refraction mechanism for the probability of “freak wave” formation in the ocean. The distribution of these anomalously tall waves (which are believed to sink one large ship as often as once a month) is predicted to depend on a “freak index”, or ratio between the average angle angle of wave deflection by current eddies to the angular spread of the incoming sea. Similar physics is believed to be relevant in optics (twinkling of starlight) and in acoustics (long-range sound transport through the ocean).

L. Kaplan, “Statistics of Branched Flow in a Weak Correlated Random Potential,” Phys. Rev. Lett. 89, 184103 (2002), arXiv:nlin.CD/0206040.

“Refraction of Gaussian Seas: Rare Event Statistics,” Workshop on Rogue Waves, International Centre for Mathematical Sciences, Edinburgh, Dec 12, 2005.

Interaction Matrix Element Fluctuations in Quantum Dots:

Nanostructures such as quantum dots can be thought of as artificial atoms, and their possible technological applications range from laser physics to solid-state quantum computation. These devices display mesoscopic, sample-specific, fluctuations based on quantum phase coherence. They are ideal laboratories for studying the interplay between chaos and electron-electron interactions.

In the Coulomb blockade regime of a ballistic quantum dot, the distribution of conductance peak spacings is well known to be incorrectly predicted by a single-particle picture; instead, matrix element fluctuations of the residual electron-electron interaction need to be taken into account. However, a simple random wave model for the Hartree-Fock wave functions, valid in the semiclassical limit where the number of electrons in the dot becomes large, predicts fluctuations that are far too small to explain the experimental data. With Yoram Alhassid of Yale University, we have examined matrix element fluctuations in realistic chaotic geometries, and shown that at energies of experimental interest these fluctuations exceed by a factor of 2 to 4 the predictions of the random wave model. The enhancements arise from terms that are formally subleading, and allow much better agreement between the Hartree-Fock picture and low-temperature experiments.

Naive semiclassical methods are typically inadequate for accurate predictions of quantities such as the distribution of interaction matrix elements, for system sizes of experimental interest. With student Matthew Smith, we are developing more sophisticated methods for making such predictions based only on short-time single-particle dynamics in a given geometry.

L. Kaplan and Y. Alhassid, “Interaction Matrix Elements in Chaotic Billiards,” in preparation.

“Interaction Matrix Element Fluctuations in Quantum Dots,” International Workshop “Correlations in quantum systems: quantum dots, quantum gases and nuclei,” Palma de Mallorca, Spain, Sep 27, 2005.

Structure of Wave Functions and Quantum Transport in Generic Systems:

Long-time and stationary quantum behavior always retains an imprint of short-time dynamics, even though such memory is absent from the long-time classical behavior. Thus, statistical properties of individual quantum wave functions in real systems may differ greatly from the predictions of random matrix theory (RMT), which contains no dynamical information and predicts Gaussian random fluctuations for the wave amplitudes. Wave function scarring, the imprint of unstable periodic orbits of the corresponding classical system, is one of the most visually striking properties of quantum chaotic behavior, and is observed in microwave cavities, semiconductor heterostructures, vibrating metal plates, and soap films, among other systems. We have obtained a fully quantitative and robust semiclassical theory for the distribution of wave function intensities in a chaotic system, depending only on the stability of the short periodic orbits. Large intensities have been observed numerically with a frequency exceeding by a factor of 10³⁷ that expected from RMT. In open systems, the probability to remain inside the system at long times, as well as conductance or reaction rate statistics, are very strongly affected by the location of an opening or lead, once again in contrast with the long-time classical behavior.

The imprint of short-time dynamics on eigenstate structure is not limited to situations where the short-time dynamics is related to classical unstable periodic orbits. For example, scar-like resonances related to diffractive trajectories have been observed in 2DEG conductance experiments. The approach of using short-time behavior to understand eigenstate structure has been used successfully by us in many situations where a proper classical limit does not exist, such as two-body random interactions in a many-fermion system or dynamics on a quantum graph or lattice, as well as in pseudo-integrable systems where the Lyapunov exponent vanishes. Short-time dynamical effects cause fine-scale wave function structure and transport in some classically ergodic systems to become less and less ergodic in the classical limit of short wavelength, i.e. each eigenstate becomes localized in an ever decreasing fraction of the available phase space.

As discussed recently by Richard Weaver (Urbana-Champaign), related techniques may be applied to predict energy transport in large structural acoustic systems.

K. Damborsky and L. Kaplan, “Scar Intensity Statistics in the Position Representation,” Phys. Rev. E 72, 066204 (2005), arXiv:nlin.CD/0510040. 

L. Kaplan, “Correlation Function Bootstrapping in Quantum Chaotic Systems,” Phys. Rev. E 71, 056212 (2005), arXiv:nlin.CD/0503058.

L. Kaplan, “Eigenstate Structure in Graphs and Disordered Lattices,” Phys. Rev. E 64, 036225 (2001), arXiv:nlin.CD/0101048. 

W. E. Bies, L. Kaplan, and E. J. Heller, “Scarring Effects on Tunneling in Chaotic Double-Well Potentials,” Phys. Rev. E 64, 016204 (2001), arXiv:nlin.CD/0007037. 

W. E. Bies, L. Kaplan, M. R. Haggerty, and E. J. Heller, “Localization of Eigenfunctions in the Stadium Billiard,” Phys. Rev. E 63, 066214 (2001), arXiv:nlin.CD/0004024. 

L. Kaplan, “Structure of Quantum Chaotic Wavefunctions: Ergodicity, Localization, and Transport,” invited article for special issue of Physica E 9, 502-8 (2001), arXiv:chao-dyn/9911003.

L. Kaplan, “Periodic Orbit Effects on Conductance Peak Heights in a Chaotic Quantum Dot,” Phys. Rev. E 62, 3476-88 (2000), arXiv:nlin.CD/0003013. 

L. Kaplan and E. J. Heller, “Short-Time Effects on Eigenstate Structure in Sinai Billiards and Related Systems,” Phys. Rev. E 62, 409-26 (2000), arXiv:chaodyn/9910030. 

L. Kaplan and E. J. Heller, “Theory of Eigenfunction Scarring,” NATO ASI series volume Supersymmetry and Trace Formulae: Chaos and Disorder, ed. by I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii (Plenum, 1999). 

L. Kaplan, “Scar and Antiscar Quantum Effects in Open Chaotic Systems,” Phys. Rev. E 59, 5325-37 (1999), arXiv:chao-dyn/9809013. 

L. Kaplan, “Recent Developments in the Theory of Scarring,” invited review article for Nonlinearity 12, R1-R40 (1999), arXiv:chao-dyn/9810013. 

L. Kaplan, “Wavefunction Intensity Statistics from Unstable Periodic Orbits,” Phys. Rev. Lett. 80, 2582-5 (1998), arXiv:chao-dyn/9809012. 

L. Kaplan and E. J. Heller, “Measuring Scars of Periodic Orbits,” Phys. Rev. E 59, 6609-28 (1999), arXiv:chao-dyn/9812011.

L. Kaplan and E. J. Heller, “Linear and Nonlinear Theory of Eigenfunction Scars,” Annals of Physics 264, 171-206 (1998), arXiv:chao-dyn/9809011. 

L. Kaplan and E. J. Heller, “Weak Quantum Ergodicity,” Physica D 121, 1-18 (1998), arXiv:chao-dyn/9810002.

Many-Body Interacting Systems:

Realistic interacting systems, including examples from mesoscopic, nuclear, and atomic physics, are not well-modeled by random matrix theory, which assumes that all many-body interactions are equally important. A much better model is a Hamiltonian consisting only of the mean field plus 2-body residual forces. In many-fermion systems with random 2-body interactions, eigenstates near the ground state are much more localized in the Fock basis than the high-lying eigenstates, i.e. they are very structured and composed of a relatively small number of non-interacting Slater determinants. The degree of localization as a function of energy is given by a simple analytic expression and is consistent with nuclear shell model calculations for realistic interactions as well as with calculations in atomic physics.

Similarly, a 2-body random interaction model for spin systems correctly predicts the observed dominance of spin 0 ground states; the spin structure of low-lying states can largely be explained analytically. Strong correlations between wave functions and energies of low-lying states of different spin indicate the presence of some degree of collective behavior. Odd-even stagger in ground state energies, which is observed for example in nuclei, metallic clusters, and small metallic grains, has also been shown to arise in a model containing only a random 2-body interaction, without the need for single-particle orbital energy effects or a coherent pairing interaction.

Overall, we find that a significant amount of spectral and wave function structure is consistent with random 2-body interactions. Many typical properties of realistic interacting systems are correctly reproduced using a completely random Hamiltonian, once the 2-body nature of the interaction is included.

Collaborators: George Bertsch (University of Washington), Thomas Papenbrock (now at Oak Ridge), and Calvin Johnson (now at San Diego State)

T. Papenbrock, L. Kaplan, and G. F. Bertsch, “Odd-even Binding Effect from Random Two-Body Interactions,” Phys. Rev. B 65, 235120 (2002), arXiv:condmat/0202493. 

L. Kaplan, T. Papenbrock, and C. W. Johnson, “Spin Structure of Many-Body Systems with Two-Body Random Interactions,” Phys. Rev. C 63, 014307 (2001), arXiv:nucl-th/0007013. 

L. Kaplan and T. Papenbrock, “Wave Function Structure in Two-Body Random Matrix Ensembles,” Phys. Rev. Lett. 84, 4553-6 (2000), arXiv:nucl-th/9911038.

Time Ordering and Time Correlation in Kicked Qubits:

Predicting the full quantum evolution of an N-body system is exponentially difficult. Widely used is the independent particle approximation (IPA), in which the interaction is averaged over, and the many-body Hamiltonian is replaced by a sum of effective 1-body mean-field Hamiltonians. The breakdown of this approximation is associated with inter-particle correlations, which are of great interest in quantum control and the study of quantum entanglement. In a similar spirit, one may study the independent time approximation (ITA), where the interaction Hamiltonian is averaged over time rather than over space. In such an approximation, multiple times may be used for multiple particles. The breakdown of this approximation is due to time-interaction effects, associated with non-commuting interactions at different times, in the intermediate (Dirac) picture of time evolution. Alternatively, time interaction may be associated with off-shell effects in the energy-domain Green's function. Time interaction effects have been observed experimentally in two-electron transitions in high-velocity atomic collisions. Closely related to the ITA is the no-time-ordering (NTO) approximation, which eliminates time ordering within single-body terms in the Hamiltonian as well as in the interactions. This is equivalent to setting Dyson's time ordering operator to unity.

Qubits, or two-level systems, 'kicked' or switched by an external field are of obvious importance in quantum information theory. Due to their analytic tractability, they provide a simple model for studying general properties of time correlation and time ordering effects in strongly interacting systems. Thus, we may better understand 'how time works' in quantum mechanics.

Colaborators include: Jim McGuire (Tulane Physics), Alex Burin (Tulane Chemistry), and Dmitry Uskov (LSU)

J. H. McGuire, L. Kaplan, Kh. Kh. Shakov, A. Chalastaras, A. M. Smith, A. Godunov, H. Schmidt-Bcking, and D. Uskov, “How Time Works in Quantum Systems: Overview of Time Ordering and Time Correlation in Weakly Perturbed Atomic Collisions and in Strongly Perturbed Qubits,” arXiv:quant-ph/0512254. 

A. Chalastaras, L. Kaplan, Kh. Kh. Shakov, M. Smith, and J. H. McGuire, “An Overview of Simply Pulsed Qubits,” arXiv:quant-ph/0507138. 

Kh. Kh. Shakov, J. H. McGuire, L. Kaplan, A. Chalastaras, and D. Uskov, “Sudden Switching in Qubits,” J. Phys. B 39, 1361 (2006), arXiv:quant-ph/0503086.

L. Kaplan, Kh. Kh. Shakov, A. Chalastaras, M. Maggio, A. L. Burin, and J. H. McGuire, “Time Ordering in Kicked Qubits,” Phys. Rev. A 70, 063401 (2004), arXiv:quant-ph/0406177, also appearing in Virtual Journal of Quantum Information and Virtual Journal of Ultrafast Science.

Long-Time Semiclassical Accuracy and Quantization Ambiguity:

Semiclassical methods provide a bridge between our classically-based intuition and the quantum world, allowing us to separate physical behavior due simply to classical paths and their interference from behavior attributable to non-classical processes such as tunneling or diffraction. They permit straightforward calculations in situations where a full quantum analysis is impractical; in other situations a full quantum analysis may be numerically feasible but will not provide the insight that emerges only from a semiclassical approach. Before confidently applying semiclassical approaches to investigate long-time transport and stationary quantum properties (including individual wave functions) in chaotic systems, we need a better understanding of the scaling of semiclassical errors at long times, including the dependence of the semiclassical error on parameters such as effective hbar, degree of chaos, and the dimensionality of the system. In the past, long-time semiclassical calculations in chaotic systems have been plagued by the exponential proliferation of classical trajectories.

One way to overcome this problem is the multiplicative approach, used by us to demonstrate that dynamical localization in quasi-one-dimensional classically diffusive systems is a semiclassical effect, requiring only constructive interference among returning classical paths without 'hard quantum' corrections. The numerical calculations extended to time scales where brute-force semiclassical calculations would require summing over 10¹⁰⁰ classical paths. More generally, the multiplicative approach can be used to show that the mean squared error in the semiclassical propagator grows only linearly with time for a classically chaotic system, in contrast with the quadratic growth that is predicted for classically regular dynamics. Thus, at long times, the semiclassical approximation is more accurate in chaotic than in regular systems. In two dimensions, individual spectral levels may be semiclassically resolved if the classical limit is chaotic, but not otherwise.

A related result is that the so-called quantization ambiguity is strongly suppressed in classically ergodic systems. Specifically, O(hbar²) ambiguities in the quantum  Hamiltonian that depend on the way in which a particle is bound to a d=2 surface prevent the construction of a quantum spectrum using only intrinsic properties of the surface if the classical motion on the surface is integrable, but both spectrum and wave functions can be unambiguously constructed in the chaotic case. The problems of semiclassical accuracy and quantization ambiguity are closely related the Loschmidt echo (loss of accuracy when quantum evolution is performed using a slightly perturbed Hamiltonian) and to the Brownian motion model of spectral equilibration in random matrix theory.

L. Kaplan, “Brownian Motion Model of Quantization Ambiguity and Semiclassical Accuracy in Chaotic Systems,” Phys. Rev. E 72, 036214 (2005), arXiv:nlin.CD/0507046.

L. Kaplan, “Semiclassical Accuracy in Phase Space for Regular and Chaotic Dynamics,” Phys. Rev. E 70, 026223 (2004), arXiv:nlin.CD/0406054.

L. Kaplan, “Quantization Ambiguity, Ergodicity, and Semiclassics,” New J. Phys. 4, 90 (2002), arXiv:quant-ph/9906065.

L. Kaplan, “Semiclassical Dynamical Localization and the Multiplicative Semiclassical Propagator,” Phys. Rev. Lett. 81, 3371-4 (1998), arXiv:chaodyn/9809007.

L. Kaplan, “Multiplicative Semiclassical Dynamics and the Quantization Time,” Phys. Rev. E 58, 2983-91 (1998), arXiv:chao-dyn/9809006.

L. Kaplan, N. Maitra, and E. J. Heller, “Quantizing Constrained Systems: New Perspectives” Phys. Rev. A 56, 2592-9 (1997), arXiv:quant-ph/9810037.

L. Kaplan and E. J. Heller, “Overcoming the Wall in the Semiclassical Baker’s map,” Phys. Rev. Lett. 76, 1453-6 (1996), arXiv:chao-dyn/9809008.

Vacuum Energy in Chaotic Geometries:

Whether and why the Casimir (vacuum energy or zero-point energy) effect is attractive or repulsive in any particular scenario is a notorious mystery. Recently increased appreciation of the connection between vacuum energy and periodic or closed classical orbits suggests that the sign is related to the phase of the oscillations of the density of eigenvalues — i.e., Maslov indices and their generalizations. Classical-path methods seem to promise accurate calculations of quantum vacuum energies in circumstances where exact solutions, or naive extrapolations (such as the proximity force approximation), are inapplicable. We are particularly interested in investigating the accuracy of classical-path approximations in closed chaotic systems, including graphs and billiards, where the vacuum energy may be alternatively computed directly from the spectrum. This work is also expected to shed light on the effect on the vacuum energy of singular boundaries, such as edges and corners, and to improve our understanding of the relationship between local vacuum energy (which in the absence of regularization becomes infinite near a boundary) and the total energy.

Our work on vacuum energy is part of a math-physics collaboration that also includes Ricardo Estrada (LSU), Stephen Fulling (Texas A&M), Klaus Kirsten (Baylor) and Kimball Milton (University of Oklahoma).

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