Quantum Chaos – Srinivas Sridhar

Quantum Chaos is the study of the quantum mechanics of classically chaotic systems. The field is currently a very active area of research. Our main thrust is to explore, via electromagnetic experiments, the manifestations of classical chaos in wave mechanics.
Our research program has led to several noteworthy observations listed below:
Experimental “proof” of a mathematical theorem on “Not Hearing the Shape of Drums”. Click here to see cover story in Science News.
Direct experimental observation of scars in quantum eigenfunctions of microwave cavities. Phys. Rev. Lett. , 67, 785 (1991)
Experimental observation of localized wavefunctions in disordered billiards, and deviations from the Porter-Thomas distribution due to localization in disordered billiards. These were the first experiments to be performed on disordered billiards. Phys. Rev. Lett., 75, 822 (1995)
First experimental observation of quantum fingerprints of classical Ruelle-Pollicott resonances. Phys. Rev. Lett., 85, 2360 (2000), Nobel Symposium on “Quantum Chaos Y2K”
Experimental studies of correlations of chaotic and disordered eigenfunctions and comparison with supersymmetry nonlinear sigma models. Phys. Rev. Lett., 85, 2360 (2000)
Tunneling Proximity Resonances: Experimental observation using dielectric resonators Phys. Lett. A, 268 (4-6), 399 (2000)
Precision tests of universal aspects of quantum spectra, observation of a “correlation hole”, and tests of applicability of Random Matrix Theories to microwave cavities. Phys. Rev. E. (Rapid Comm.), 49, R11 (1994)
Observation of Porter-Thomas distribution and fluctuations in eigenfunctions of chaotic billiards. Phys. Rev. Lett., 75, 822 (1995)
Observation of quantum resonances, universal properties, and comparison with semiclassical theories of the n-disk system. Phys. Rev. Lett., 82, 5233 (1999), Phys. Rev. E, 61, 3652 (2000)
It is evident that the experiments are able to explore a remarkable range of issues in Quantum Chaos, including tests of fundamental theories and different classical limits. The microwave experiments have yielded entirely new insights and perspectives concerning the quantum-classical correspondence. The reasons for this success are the ability to study well-defined geometries where the classical dynamics is clearly known, and the precision and flexibility of the experiments. The experiments have led to tests of key theoretical ideas while at the same time raising entirely new questions and motivating theorists in new directions.
Our work in this area is currently funded by National Science Foundation (Division of Atomic, Molecular and Optical Physics).
Localization of Wavefunctions in Disordered Billiards

Experimental observation of localized wavefunctions in disordered billiards, and deviations from the Porter-Thomas distribution due to localization in disordered billiards. These were the first experiments to be performed on disordered billiards.
We show the evolution of localization with increasing scattering by measuring wavefunctions in disordered billiards.
Chemical & Biological Physics Collective Excitations Education Experimental Mathematics featured Magnetism Metamaterials and Nanophotonics MRI Nanomaterials Nanomedicine Neurotechnology Opthalmology Quantum Chaos Research Policy Sensors Superconductivity 
 Lu, WT;  Zeng, Weiqiao;  Sridhar, S
Duality between quantum and classical dynamics for integrable billiards Journal Article 
In: Physical Review E, vol. 73, no. 4, pp. 046201, 2006.
Abstract | Tags: Quantum Chaos
@article{lu2006duality,

title = {Duality between quantum and classical dynamics for integrable billiards},

author = {WT Lu and Weiqiao Zeng and S Sridhar},

year  = {2006},

date = {2006-01-01},

journal = {Physical Review E},

volume = {73},

number = {4},

pages = {046201},

publisher = {APS},

abstract = {We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the quantum wave vector spectrum. A semiclassical derivation and numerical calculations are presented in support of the results. These classical eigenmodes can be observed in physical experiments through the autocorrelation of the transmission coefficient of waves in quantum billiards. Exact classical trace formulas of the resolvent are derived for the rectangle, equilateral triangle, and circle billiards. We also establish a correspondence between the classical periodic orbit length spectrum and the quantum spectrum for integrable polygonal billiards.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the quantum wave vector spectrum. A semiclassical derivation and numerical calculations are presented in support of the results. These classical eigenmodes can be observed in physical experiments through the autocorrelation of the transmission coefficient of waves in quantum billiards. Exact classical trace formulas of the resolvent are derived for the rectangle, equilateral triangle, and circle billiards. We also establish a correspondence between the classical periodic orbit length spectrum and the quantum spectrum for integrable polygonal billiards.
Close
 Lu, WT;  Sridhar, S
Long range correlations among the eigenvalues of polygonal billiards and Riemann zeros Journal Article 
In: APS, vol. 2004, pp. D19–004, 2004.
Abstract | Tags: Quantum Chaos
@article{lu2004long,

title = {Long range correlations among the eigenvalues of polygonal billiards and Riemann zeros},

author = {WT Lu and S Sridhar},

year  = {2004},

date = {2004-01-01},

journal = {APS},

volume = {2004},

pages = {D19--004},

abstract = {The momentum eigenvalue spectrum of polygonal billiards is shown to display a remarkable invariance when analyzed in terms of the spectral auto-correlation. The auto-correlation of any spectral window is encoded with the lowest eigenvalues. Thus a resurgence of the lowlying eigenstates occurs throughout the entire spectrum. We show that this can be understood in terms of the trace formula if the periodic orbits are stable, as occurs in integrable and pseudo-integrable polygonal billiards. The same resurgence also occurs in the nontrivial zeros of the Riemann zeta function, albeit with negative amplitude. The analytical arguments are supported by numerical calculations. Work supported by NSF.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
The momentum eigenvalue spectrum of polygonal billiards is shown to display a remarkable invariance when analyzed in terms of the spectral auto-correlation. The auto-correlation of any spectral window is encoded with the lowest eigenvalues. Thus a resurgence of the lowlying eigenstates occurs throughout the entire spectrum. We show that this can be understood in terms of the trace formula if the periodic orbits are stable, as occurs in integrable and pseudo-integrable polygonal billiards. The same resurgence also occurs in the nontrivial zeros of the Riemann zeta function, albeit with negative amplitude. The analytical arguments are supported by numerical calculations. Work supported by NSF.
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 Lu, WT;  Sridhar, S
Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality Journal Article 
In: arXiv preprint nlin/0405058, 2004.
Abstract | Tags: Quantum Chaos
@article{lu2004correlations,

title = {Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality},

author = {WT Lu and S Sridhar},

year  = {2004},

date = {2004-01-01},

journal = {arXiv preprint nlin/0405058},

abstract = {We present a conjecture describing new long range correlations among the Riemann zeros leading to 3 principal features:(i) The spectral auto-correlation is invariant wrt the averaging window.(ii) Resurgence occurs wherein the lowest zeros appear in all auto-correlations.(iii) Suitably defined correlations lead to predictions (prophecy) of new zeros. This conjecture is supported by analytical arguments and confirmed by numerical calculations using 10^{22} zeros computed by Odlyzko. The results lead to a self-duality of the Riemann spectrum similar to the quantum-classical duality observed in billiards.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We present a conjecture describing new long range correlations among the Riemann zeros leading to 3 principal features:(i) The spectral auto-correlation is invariant wrt the averaging window.(ii) Resurgence occurs wherein the lowest zeros appear in all auto-correlations.(iii) Suitably defined correlations lead to predictions (prophecy) of new zeros. This conjecture is supported by analytical arguments and confirmed by numerical calculations using 10^{22} zeros computed by Odlyzko. The results lead to a self-duality of the Riemann spectrum similar to the quantum-classical duality observed in billiards.
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 Lu, WT;  Sokoloff, JB;  Sridhar, S
Classical physics, including nonlinear media and photonic materials-Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative Journal Article 
In: Physical Review-Section E-Statistical Nonlinear and Soft Matter Physics, vol. 69, no. 2, pp. 26604–26604, 2004.
Tags: Quantum Chaos
@article{lu2004classical,

title = {Classical physics, including nonlinear media and photonic materials-Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative},

author = {WT Lu and JB Sokoloff and S Sridhar},

year  = {2004},

date = {2004-01-01},

journal = {Physical Review-Section E-Statistical Nonlinear and Soft Matter Physics},

volume = {69},

number = {2},

pages = {26604--26604},

publisher = {Melville, NY: Published by the American Physical Society through the~…},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
 Wen-tao, Lu T;  Sridhar, Srinivas;  Zworski, Maciej
Fractal Weyl laws for chaotic open systems Journal Article 
In: 2003.
Abstract | Tags: Quantum Chaos
@article{wenfractal,

title = {Fractal Weyl laws for chaotic open systems},

author = {Lu T Wen-tao and Srinivas Sridhar and Maciej Zworski},

year  = {2003},

date = {2003-10-08},

abstract = {We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.
Close
 Pradhan, Prabhakar;  Lu, Wentao T;  Sridhar, S
Statistics of the Eigenfunctions of Chaotic and Disordered Quantum Systems: A Disordered Tight Binding Model Calculation Proceedings Article 
In: APS Meeting Abstracts, 2003.
Abstract | Tags: Quantum Chaos
@inproceedings{pradhan2003statistics,

title = {Statistics of the Eigenfunctions of Chaotic and Disordered Quantum Systems: A Disordered Tight Binding Model Calculation},

author = {Prabhakar Pradhan and Wentao T Lu and S Sridhar},

year  = {2003},

date = {2003-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {We analyze the statistical properties of the eigenfunctions of the Anderson disordered tight binding Hamiltonian, for an electron in closed 2-D, chaotic and disordered systems. For chaotic systems, the inverse participation ratio (IPR) that measures the amount of localization of an eigenstate, has a narrow and symmetric distribution peaked around IPR = 3, as predicted by random matrix theory. For disordered systems, the distribution is asymmetric and peaks at IPR > 3. As a function of energy, the IPR distribution decays as a power law with exponent 1/2 at low energies (i.e. more localized state), and saturates at IPR =3 (delocalized state) for larger energies. The spatial intensity auto-correlations of the eigenfunctions are strong for a more localized state at short distances, and they decay via a Friedel oscillation, as predicted by nonlinear sigma models, with a decay length scale corresponding to the localization length. For weak to moderate disorder in 2D, our numerical calculations are consistent with the random matrix theory, non-linear sigma models, and also with our previous experimental results for 2D quantum chaotic and disordered systems.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
We analyze the statistical properties of the eigenfunctions of the Anderson disordered tight binding Hamiltonian, for an electron in closed 2-D, chaotic and disordered systems. For chaotic systems, the inverse participation ratio (IPR) that measures the amount of localization of an eigenstate, has a narrow and symmetric distribution peaked around IPR = 3, as predicted by random matrix theory. For disordered systems, the distribution is asymmetric and peaks at IPR > 3. As a function of energy, the IPR distribution decays as a power law with exponent 1/2 at low energies (i.e. more localized state), and saturates at IPR =3 (delocalized state) for larger energies. The spatial intensity auto-correlations of the eigenfunctions are strong for a more localized state at short distances, and they decay via a Friedel oscillation, as predicted by nonlinear sigma models, with a decay length scale corresponding to the localization length. For weak to moderate disorder in 2D, our numerical calculations are consistent with the random matrix theory, non-linear sigma models, and also with our previous experimental results for 2D quantum chaotic and disordered systems.
Close
 Lu, WT;  Sridhar, Srinivas;  Zworski, Maciej
Fractal Weyl laws for chaotic open systems Journal Article 
In: Physical review letters, vol. 91, no. 15, pp. 154101, 2003.
Abstract | Tags: Quantum Chaos
@article{lu2003fractal,

title = {Fractal Weyl laws for chaotic open systems},

author = {WT Lu and Srinivas Sridhar and Maciej Zworski},

year  = {2003},

date = {2003-01-01},

journal = {Physical review letters},

volume = {91},

number = {15},

pages = {154101},

publisher = {APS},

abstract = {We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.
Close
 Sato, Daisuke;  Lu, Wentao;  Pradhan, Prabhakar;  Sridhar, Srinivas
Spectral statistics of microwave disordered billiards Proceedings Article 
In: APS Meeting Abstracts, 2003.
Abstract | Tags: Quantum Chaos
@inproceedings{sato2003spectral,

title = {Spectral statistics of microwave disordered billiards},

author = {Daisuke Sato and Wentao Lu and Prabhakar Pradhan and Srinivas Sridhar},

year  = {2003},

date = {2003-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {Using a new algorithm to analyze experimental microwave spectra, we are able to accurately obtain nearly 1000 eigenvalues for microwave disordered billiards with no missing energy level. The cumulative level density N (E) of all billiards is in good agreement with the Weyl formula. We also determine statistical measures such as P (s), Delta_3, R_2, Y_2, Sigma_2, and study their dependence on the mean free path and localization length. The experimental results are compared with the supersymmetry sigma model and tight binding calculations. Work supported by NSF-0098801.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
Using a new algorithm to analyze experimental microwave spectra, we are able to accurately obtain nearly 1000 eigenvalues for microwave disordered billiards with no missing energy level. The cumulative level density N (E) of all billiards is in good agreement with the Weyl formula. We also determine statistical measures such as P (s), Delta_3, R_2, Y_2, Sigma_2, and study their dependence on the mean free path and localization length. The experimental results are compared with the supersymmetry sigma model and tight binding calculations. Work supported by NSF-0098801.
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 Dhar, Abhishek;  Rao, Madhusudhana D;  Sridhar, S;  others,
Isospectrality in chaotic billiards Journal Article 
In: Physical Review E, vol. 68, no. 2, pp. 026208, 2003.
Abstract | Tags: Quantum Chaos
@article{dhar2003isospectrality,

title = {Isospectrality in chaotic billiards},

author = {Abhishek Dhar and Madhusudhana D Rao and S Sridhar and others},

year  = {2003},

date = {2003-01-01},

journal = {Physical Review E},

volume = {68},

number = {2},

pages = {026208},

publisher = {APS},

abstract = {We consider a modification of isospectral cavities whereby the classical dynamics changes from pseudointegrable to chaotic. We construct an example where we can prove that isospectrality is retained. We then demonstrate this explicitly in microwave resonators.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We consider a modification of isospectral cavities whereby the classical dynamics changes from pseudointegrable to chaotic. We construct an example where we can prove that isospectrality is retained. We then demonstrate this explicitly in microwave resonators.
Close
 Sridhar, S;  Lu, WT
Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave experiments Journal Article 
In: Journal of statistical physics, vol. 108, no. 5-6, pp. 755–765, 2002.
Abstract | Tags: Quantum Chaos
@article{sridhar2002sinai,

title = {Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave experiments},

author = {S Sridhar and WT Lu},

year  = {2002},

date = {2002-01-01},

journal = {Journal of statistical physics},

volume = {108},

number = {5-6},

pages = {755--765},

publisher = {Kluwer Academic Publishers-Plenum Publishers},

abstract = {We discuss the impact of recent developments in the theory of chaotic dynamical systems, particularly the results of Sinai and Ruelle, on microwave experiments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as “scars” in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We discuss the impact of recent developments in the theory of chaotic dynamical systems, particularly the results of Sinai and Ruelle, on microwave experiments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as “scars” in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.
Close
 Pradhan, Prabhakar;  Sridhar, S
From chaos to disorder: Statistics of the eigenfunctions of microwave cavities Journal Article 
In: Pramana, vol. 58, no. 2, pp. 333–341, 2002.
Abstract | Tags: Quantum Chaos
@article{pradhan2002chaos,

title = {From chaos to disorder: Statistics of the eigenfunctions of microwave cavities},

author = {Prabhakar Pradhan and S Sridhar},

year  = {2002},

date = {2002-01-01},

journal = {Pramana},

volume = {58},

number = {2},

pages = {333--341},

publisher = {Springer India},

abstract = {We study the statistics of the experimental eigenfunctions of chaotic and disordered microwave billiards in terms of the moments of their spatial distributions, such as the inverse participation ratio (IPR) and density-density auto-correlation. A path from chaos to disorder is described in terms of increasing IPR. In the chaotic, ballistic limit, the data correspond well with universal results from random matrix theory. Deviations from universal distributions are observed due to disorder induced localization, and for the weakly disordered case the data are well-described by including finite conductance and mean free path contributions in the framework of nonlinear sigma models of supersymmetry.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We study the statistics of the experimental eigenfunctions of chaotic and disordered microwave billiards in terms of the moments of their spatial distributions, such as the inverse participation ratio (IPR) and density-density auto-correlation. A path from chaos to disorder is described in terms of increasing IPR. In the chaotic, ballistic limit, the data correspond well with universal results from random matrix theory. Deviations from universal distributions are observed due to disorder induced localization, and for the weakly disordered case the data are well-described by including finite conductance and mean free path contributions in the framework of nonlinear sigma models of supersymmetry.
Close
 Sridhar, S;  Lu, WT
Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave experiments. e-print Journal Article 
In: 2002.
Abstract | Tags: Quantum Chaos
@article{sridhar2002sinaib,

title = {Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave experiments. e-print},

author = {S Sridhar and WT Lu},

year  = {2002},

date = {2002-01-01},

abstract = {We discuss the impact of recent developments in the theory of chaotic dynami-cal systems, particularly the results of Sinai and Ruelle, on microwave experi-ments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as ‘‘scars’’in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We discuss the impact of recent developments in the theory of chaotic dynami-cal systems, particularly the results of Sinai and Ruelle, on microwave experi-ments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as ‘‘scars’’in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.
Close
 Sinai, Ya G;  Young, Lai-Sang;  Sridhar, S;  Lu, WT; van Beijeren, Henk;  Dorfman, JR;  Jakšic, V;  Pillet, CA;  Gallavotti, G;  Lebowitz, JL;  others,
Some Ill-Formulated Problems on Regular and Messy Behavior in Statis-tical Mechanics and Smooth Dynamics for Which I Would Like the Advice of Yasha Sinai David Ruelle What, in My Opinion, David Ruelle Should Do in the Coming Years Journal Article 
In: Journal of Statistical Physics, vol. 108, no. 3/4, 2002.
Tags: Quantum Chaos
@article{sinai2002some,

title = {Some Ill-Formulated Problems on Regular and Messy Behavior in Statis-tical Mechanics and Smooth Dynamics for Which I Would Like the Advice of Yasha Sinai David Ruelle What, in My Opinion, David Ruelle Should Do in the Coming Years},

author = {Ya G Sinai and Lai-Sang Young and S Sridhar and WT Lu and Henk van Beijeren and JR Dorfman and V Jakšic and CA Pillet and G Gallavotti and JL Lebowitz and others},

year  = {2002},

date = {2002-01-01},

journal = {Journal of Statistical Physics},

volume = {108},

number = {3/4},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

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 Lu, WT;  Sridhar, S;  others,
Spectra and wave functions of open chaotic billiards Proceedings Article 
In: APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts, 2002.
Abstract | Tags: Quantum Chaos
@inproceedings{lu2002spectra,

title = {Spectra and wave functions of open chaotic billiards},

author = {WT Lu and S Sridhar and others},

year  = {2002},

date = {2002-01-01},

booktitle = {APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts},

abstract = {The quantum spectra and wave functions of the n-disk open chaotic system are studied. The spectra consist of quantum resonances which are calculated in period orbit theory and measured in microwave experiments. The correlation of quantum resonances leads to the classical Ruelle-Pollicott resonances. The scattering wave functions are studied numerically and experimentally. Scars are observed and wave function statistics are analyzed. Spectra and wave functions of divided phase space billiard are also studied. Work supported by NSF-PHY-0098801},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
The quantum spectra and wave functions of the n-disk open chaotic system are studied. The spectra consist of quantum resonances which are calculated in period orbit theory and measured in microwave experiments. The correlation of quantum resonances leads to the classical Ruelle-Pollicott resonances. The scattering wave functions are studied numerically and experimentally. Scars are observed and wave function statistics are analyzed. Spectra and wave functions of divided phase space billiard are also studied. Work supported by NSF-PHY-0098801
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 Sridhar, Srinivas
Quantum Chaos in Microwave Billiards Proceedings Article 
In: APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts, 2002.
Abstract | Tags: Quantum Chaos
@inproceedings{sridhar2002quantum,

title = {Quantum Chaos in Microwave Billiards},

author = {Srinivas Sridhar},

year  = {2002},

date = {2002-01-01},

booktitle = {APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts},

abstract = {I discuss some recent themes from microwave experiments designed to explore issues in Quantum Chaos. The experiments measure spectra and eigenfunctions of model geometries in the form of closed and open billiards. The microwave billiards provide a nearly ideal laboratory realization of a particle in hard-wall 2-D potentials, suitable for exploring the quantum-classical correspondence in chaotic systems, and capture the essential features of diverse situations in atomic and optical physics. The experiments reveal universal features of spectral and eigenfunction statistics that are well described by random matrix theory. Non-universal features are also observed, particularly periodic orbit contributions such as scars in eigenfunctions. A systematic trend from chaos to disorder is studied in disordered billiards, where quantum diffusion and interference lead to localization and non-universal behavior of density …

},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
I discuss some recent themes from microwave experiments designed to explore issues in Quantum Chaos. The experiments measure spectra and eigenfunctions of model geometries in the form of closed and open billiards. The microwave billiards provide a nearly ideal laboratory realization of a particle in hard-wall 2-D potentials, suitable for exploring the quantum-classical correspondence in chaotic systems, and capture the essential features of diverse situations in atomic and optical physics. The experiments reveal universal features of spectral and eigenfunction statistics that are well described by random matrix theory. Non-universal features are also observed, particularly periodic orbit contributions such as scars in eigenfunctions. A systematic trend from chaos to disorder is studied in disordered billiards, where quantum diffusion and interference lead to localization and non-universal behavior of density …

Close
 Lu, Wentao T;  Prance, Kristi;  Pradhan, Prabhakar;  Sridhar, S
Quantum correlations and classical resonances in an open chaotic system Journal Article 
In: Physica Scripta, vol. 2001, no. T90, pp. 238, 2001.
Abstract | Tags: Quantum Chaos
@article{lu2001quantum,

title = {Quantum correlations and classical resonances in an open chaotic system},

author = {Wentao T Lu and Kristi Prance and Prabhakar Pradhan and S Sridhar},

year  = {2001},

date = {2001-01-01},

journal = {Physica Scripta},

volume = {2001},

number = {T90},

pages = {238},

publisher = {IOP Publishing},

abstract = {We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical Ruelle-Pollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C (κ) from the experimental quantum spectra S 21 (k). The correspondence is also established via" numerical experiments" that simulate S 21 (k) and C (κ) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open system.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical Ruelle-Pollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C (κ) from the experimental quantum spectra S 21 (k). The correspondence is also established via" numerical experiments" that simulate S 21 (k) and C (κ) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open system.
Close
 Sridhar, S
Quantum resonances and decay of a fractal repeller observed using microwaves Proceedings Article 
In: APS Meeting Abstracts, 2000.
Abstract | Tags: Quantum Chaos
@inproceedings{sridhar2000quantum,

title = {Quantum resonances and decay of a fractal repeller observed using microwaves},

author = {S Sridhar},

year  = {2000},

date = {2000-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {We describe an experimental realization of the well-known problem of n-disk scattering, which may be regarded as the "hydrogen atom" of chaotic scattering. This model geometry is related to diverse areas such as open systems, semiconductor microstructures and photodissociation. In the experiment, the quantum resonances of classically chaotic n-disk geometries were studied utilizing thin 2D microwave geometries. The transmission spectrum probes the stationary Green's function of the system, and yields both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. In the microwave experiments the wave vector correlations can be directly studied, and hence these are an interesting complement to ballistic transport in semiconductor microstructures in which correlations in the magneto-transport are obtained. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. For intermediate energies, non-universal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits. W.-T. Lu, M. Rose, K. Pance and S.Sridhar, Phys. Rev. Lett., vol 82, pp. 5233 (1999). W.-T. Lu, L. Viola, K.Pance, M. Rose and S.Sridhar, Phys. Rev. E (submitted)},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
We describe an experimental realization of the well-known problem of n-disk scattering, which may be regarded as the "hydrogen atom" of chaotic scattering. This model geometry is related to diverse areas such as open systems, semiconductor microstructures and photodissociation. In the experiment, the quantum resonances of classically chaotic n-disk geometries were studied utilizing thin 2D microwave geometries. The transmission spectrum probes the stationary Green's function of the system, and yields both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. In the microwave experiments the wave vector correlations can be directly studied, and hence these are an interesting complement to ballistic transport in semiconductor microstructures in which correlations in the magneto-transport are obtained. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. For intermediate energies, non-universal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits. W.-T. Lu, M. Rose, K. Pance and S.Sridhar, Phys. Rev. Lett., vol 82, pp. 5233 (1999). W.-T. Lu, L. Viola, K.Pance, M. Rose and S.Sridhar, Phys. Rev. E (submitted)
Close
 Sridhar, S
Quantum chaos, localization and tunnelling: microwave experiments on model geometries Journal Article 
In: Philosophical Magazine B, vol. 80, no. 12, pp. 2129–2141, 2000.
Abstract | Tags: Quantum Chaos
@article{sridhar2000quantumb,

title = {Quantum chaos, localization and tunnelling: microwave experiments on model geometries},

author = {S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Philosophical Magazine B},

volume = {80},

number = {12},

pages = {2129--2141},

publisher = {Taylor & Francis},

abstract = {Microwave experiments using two-dimensional billiard geometries are a precise test of basic issues in quantum chaos, localization and tunnelling. In closed chaotic geometries, analysis of eigenvalue statistics yields good agreement with random-matrix theory. A unique aspect of the experiments is the ability to measure eigenfunctions directly. The influence of periodic orbit scarring in chaotic eigenfunctions is directly demonstrated. Disordered microwave billiards are a textbook model system for studying the quantum properties of a single particle in a disordered potential. Localization is directly observed in eigenfunctions of the disordered billiards. Statistical properties of disordered eigenfunctions deviate from universal behaviour due to localization. These statistical properties are in good agreement with predictions from nonlinear-sigma models, although many challenges for further theoretical understanding remain. The experiments can also probe open systems, in terms of the quantum resonances and escape rate of a fractal repeller.



},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
Microwave experiments using two-dimensional billiard geometries are a precise test of basic issues in quantum chaos, localization and tunnelling. In closed chaotic geometries, analysis of eigenvalue statistics yields good agreement with random-matrix theory. A unique aspect of the experiments is the ability to measure eigenfunctions directly. The influence of periodic orbit scarring in chaotic eigenfunctions is directly demonstrated. Disordered microwave billiards are a textbook model system for studying the quantum properties of a single particle in a disordered potential. Localization is directly observed in eigenfunctions of the disordered billiards. Statistical properties of disordered eigenfunctions deviate from universal behaviour due to localization. These statistical properties are in good agreement with predictions from nonlinear-sigma models, although many challenges for further theoretical understanding remain. The experiments can also probe open systems, in terms of the quantum resonances and escape rate of a fractal repeller.



Close
 Pance, Kristi;  Lu, Wentao;  Sridhar, S
Quantum fingerprints of classical ruelle-pollicott resonances Journal Article 
In: Physical review letters, vol. 85, no. 13, pp. 2737, 2000.
Abstract | Tags: Quantum Chaos
@article{pance2000quantum,

title = {Quantum fingerprints of classical ruelle-pollicott resonances},

author = {Kristi Pance and Wentao Lu and S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Physical review letters},

volume = {85},

number = {13},

pages = {2737},

publisher = {American Physical Society},

abstract = {Quantum and classical correlations are studied experimentally in model n-disk microwave billiards. The wave vector κ autocorrelation C (κ) of the quantum spectrum displays nonuniversal oscillations for large κ, comparable to the universal random matrix theory behavior observed for small κ. The nonuniversal behavior is shown to be completely determined by the classical Ruelle-Pollicott resonances, arising from the complex eigenvalues of the Perron-Frobenius operator, and calculated using periodic orbit theory. This work establishes a fundamental connection between the quantum and classical correlations of an open system.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
Quantum and classical correlations are studied experimentally in model n-disk microwave billiards. The wave vector κ autocorrelation C (κ) of the quantum spectrum displays nonuniversal oscillations for large κ, comparable to the universal random matrix theory behavior observed for small κ. The nonuniversal behavior is shown to be completely determined by the classical Ruelle-Pollicott resonances, arising from the complex eigenvalues of the Perron-Frobenius operator, and calculated using periodic orbit theory. This work establishes a fundamental connection between the quantum and classical correlations of an open system.
Close
 Pradhan, Prabhakar;  Sridhar, S
Correlations due to localization in quantum eigenfunctions of disordered microwave cavities Journal Article 
In: Physical review letters, vol. 85, no. 11, pp. 2360, 2000.
Abstract | Tags: Quantum Chaos
@article{pradhan2000correlations,

title = {Correlations due to localization in quantum eigenfunctions of disordered microwave cavities},

author = {Prabhakar Pradhan and S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Physical review letters},

volume = {85},

number = {11},

pages = {2360},

publisher = {APS},

abstract = {Statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities are shown to demonstrate nonuniversal correlations due to localization. Varying energy E and mean free path l enable us to experimentally tune from localized to delocalized states. Large level-to-level inverse participation ratio (I 2) fluctuations are observed for the disordered billiards, whose distribution is strongly asymmetric about< I 2>. The spatial density autocorrelations of eigenfunctions are shown to spatially decay exponentially and the decay lengths are experimentally determined. All the results are quantitatively consistent with calculations based upon nonlinear sigma models.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
Statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities are shown to demonstrate nonuniversal correlations due to localization. Varying energy E and mean free path l enable us to experimentally tune from localized to delocalized states. Large level-to-level inverse participation ratio (I 2) fluctuations are observed for the disordered billiards, whose distribution is strongly asymmetric about< I 2>. The spatial density autocorrelations of eigenfunctions are shown to spatially decay exponentially and the decay lengths are experimentally determined. All the results are quantitatively consistent with calculations based upon nonlinear sigma models.
Close
 Pradhan, Prabhakar;  Pance, Kristi;  Rose, Michael;  Sridhar, S
From Localization to Chaos in experimental eigenfunctions of disordered microwave cavities Proceedings Article 
In: APS Meeting Abstracts, 2000.
Abstract | Tags: Quantum Chaos
@inproceedings{pradhan2000localization,

title = {From Localization to Chaos in experimental eigenfunctions of disordered microwave cavities},

author = {Prabhakar Pradhan and Kristi Pance and Michael Rose and S Sridhar},

year  = {2000},

date = {2000-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {We analyze eigenfunctions of disordered billiards obtained experimentally using microwave cavities, in terms of density correlations and moment distributions. Deviations from universal distributions are observed due to disorder induced localization. Level-to-level Inverse Participation Ratio (IPR I_2) fluctuations are observed and analyzed. Varying frequency (f) enables us to experimentally tune from localized to delocalized states, and this path for I_2 (f) follows a power law decay with exponent 1/2 above the cutoff frequency. In chaotic billiards, I 2 has a mean value close to that of the universal 2-dimensional (2D) limiting value of 3.0, with small level-to-level fluctuations resulting in a symmetric distribution about< I_2>. In disordered billiards not only is the mean value of I_2>> 3.0, but the fluctuations are also much greater. The IPR distribution for the disordered billiards is asymmetric about< I_2>, and is quantitatively …},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
We analyze eigenfunctions of disordered billiards obtained experimentally using microwave cavities, in terms of density correlations and moment distributions. Deviations from universal distributions are observed due to disorder induced localization. Level-to-level Inverse Participation Ratio (IPR I_2) fluctuations are observed and analyzed. Varying frequency (f) enables us to experimentally tune from localized to delocalized states, and this path for I_2 (f) follows a power law decay with exponent 1/2 above the cutoff frequency. In chaotic billiards, I 2 has a mean value close to that of the universal 2-dimensional (2D) limiting value of 3.0, with small level-to-level fluctuations resulting in a symmetric distribution about< I_2>. In disordered billiards not only is the mean value of I_2>> 3.0, but the fluctuations are also much greater. The IPR distribution for the disordered billiards is asymmetric about< I_2>, and is quantitatively …
Close
 Lu, Wentao;  Viola, Lorenza;  Pance, Kristi;  Rose, Michael;  Sridhar, S
Microwave study of quantum n-disk scattering Journal Article 
In: Physical Review E, vol. 61, no. 4, pp. 3652, 2000.
Abstract | Tags: Quantum Chaos
@article{lu2000microwave,

title = {Microwave study of quantum n-disk scattering},

author = {Wentao Lu and Lorenza Viola and Kristi Pance and Michael Rose and S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Physical Review E},

volume = {61},

number = {4},

pages = {3652},

publisher = {APS},

abstract = {We describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green’s function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green’s function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.
Close
 Pance, Kristi;  Viola, Lorenza;  Sridhar, S
Tunneling proximity resonances: interplay between symmetry and dissipation Journal Article 
In: Physics Letters A, vol. 268, no. 4-6, pp. 399–405, 2000.
Abstract | Tags: Quantum Chaos
@article{pance2000tunneling,

title = {Tunneling proximity resonances: interplay between symmetry and dissipation},

author = {Kristi Pance and Lorenza Viola and S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Physics Letters A},

volume = {268},

number = {4-6},

pages = {399--405},

publisher = {North-Holland},

abstract = {We report the first observation of bound-state proximity resonances in coupled dielectric resonators. The proximity resonances arise from the combined action of symmetry and dissipation. We argue that the large ratio between the widths is a distinctive signature of the multidimensional nature of the system. Our experiments shed light on the properties of 2D tunneling in the presence of a dissipative environment.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We report the first observation of bound-state proximity resonances in coupled dielectric resonators. The proximity resonances arise from the combined action of symmetry and dissipation. We argue that the large ratio between the widths is a distinctive signature of the multidimensional nature of the system. Our experiments shed light on the properties of 2D tunneling in the presence of a dissipative environment.
Close
 Lu, Wentao;  Viola, Lorenza;  Pance, Kristi;  Rose, Michael;  Sridhar, S
Erratum: Microwave study of quantum n-disk scattering [Phys. Rev. E 61, 3652 (2000)] Journal Article 
In: Physical Review E, vol. 62, no. 3, pp. 4478, 2000.
Abstract | Tags: Quantum Chaos
@article{lu2000erratum,

title = {Erratum: Microwave study of quantum n-disk scattering [Phys. Rev. E 61, 3652 (2000)]},

author = {Wentao Lu and Lorenza Viola and Kristi Pance and Michael Rose and S Sridhar},

year  = {2000},

date = {2000-01-01},

journal = {Physical Review E},

volume = {62},

number = {3},

pages = {4478},

publisher = {APS},

abstract = {ISSN 2470-0053 (online), 2470-0045 (print). ©2019 American Physical Society. All rights reserved. Physical Review E™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. The APS Physics logo and Physics logo are trademarks of the American Physical Society. Information about registration may be found here. Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
ISSN 2470-0053 (online), 2470-0045 (print). ©2019 American Physical Society. All rights reserved. Physical Review E™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. The APS Physics logo and Physics logo are trademarks of the American Physical Society. Information about registration may be found here. Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement.
Close
 Lu, Wentao;  Rose, M;  Pance, K;  Sridhar, S
Quantum resonances and decay of a chaotic fractal repeller observed using microwaves Journal Article 
In: Physical review letters, vol. 82, no. 26, pp. 5233, 1999.
Abstract | Tags: Quantum Chaos
@article{lu1999quantum,

title = {Quantum resonances and decay of a chaotic fractal repeller observed using microwaves},

author = {Wentao Lu and M Rose and K Pance and S Sridhar},

year  = {1999},

date = {1999-01-01},

journal = {Physical review letters},

volume = {82},

number = {26},

pages = {5233},

publisher = {APS},

abstract = {The quantum resonances of classically chaotic n-disk geometries were studied experimentally utilizing thin 2D microwave geometries. The experiments yield the frequencies and widths of low-lying resonances, which are compared with semiclassical calculations. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows nonuniversal oscillations determined by periodic orbits.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
The quantum resonances of classically chaotic n-disk geometries were studied experimentally utilizing thin 2D microwave geometries. The experiments yield the frequencies and widths of low-lying resonances, which are compared with semiclassical calculations. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows nonuniversal oscillations determined by periodic orbits.
Close
 Sridhar, S
Wave Chaos and Localization in Closed Billiards and Open Scattering Systems: Microwave Experiments Proceedings Article 
In: APS March Meeting Abstracts, 1998.
Abstract | Tags: Quantum Chaos
@inproceedings{sridhar1998wave,

title = {Wave Chaos and Localization in Closed Billiards and Open Scattering Systems: Microwave Experiments},

author = {S Sridhar},

year  = {1998},

date = {1998-01-01},

booktitle = {APS March Meeting Abstracts},

abstract = {Microwave experiments are described, which are designed to study the signatures of chaos and localization on the quantum properties of model 2-D closed (billiard) and open scattering geometries. A special advantage of the experiments is the ability to directly measure eigenfunctions. In chaotic billiards, the eigenfunctions display universal density distributions and density autocorrelations, in agreement with expressions derived from random matrix theory and from a 0D nonlinear sigma model of supersymmetry. In contrast, wavefunctions in disordered billiards show deviations from universality due to Anderson localization. The systematics of the distribution functions and inverse participation ratios are studied as a function of frequency and localization length. While results in the regime of incipient localization appear to be successfully described by leading expansions of nonlinear sigma models of supersymmetry …},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
Microwave experiments are described, which are designed to study the signatures of chaos and localization on the quantum properties of model 2-D closed (billiard) and open scattering geometries. A special advantage of the experiments is the ability to directly measure eigenfunctions. In chaotic billiards, the eigenfunctions display universal density distributions and density autocorrelations, in agreement with expressions derived from random matrix theory and from a 0D nonlinear sigma model of supersymmetry. In contrast, wavefunctions in disordered billiards show deviations from universality due to Anderson localization. The systematics of the distribution functions and inverse participation ratios are studied as a function of frequency and localization length. While results in the regime of incipient localization appear to be successfully described by leading expansions of nonlinear sigma models of supersymmetry …
Close
 Kudrolli, A;  Sridhar, S
Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard Journal Article 
In: Pramana, vol. 48, no. 2, pp. 459–467, 1997.
Abstract | Tags: Quantum Chaos
@article{kudrolli1997experiments,

title = {Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard},

author = {A Kudrolli and S Sridhar},

year  = {1997},

date = {1997-01-01},

journal = {Pramana},

volume = {48},

number = {2},

pages = {459--467},

publisher = {Springer India},

abstract = {We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.
Close
 Jacobs, T;  Willemsen, Balam A;  Sridhar, S
Quantitative analysis of nonlinear microwave surface impedance from non-Lorentzian resonances of high Q resonators Journal Article 
In: Review of scientific instruments, vol. 67, no. 10, pp. 3757–3758, 1996.
Tags: Quantum Chaos
@article{jacobs1996quantitative,

title = {Quantitative analysis of nonlinear microwave surface impedance from non-Lorentzian resonances of high Q resonators},

author = {T Jacobs and Balam A Willemsen and S Sridhar},

year  = {1996},

date = {1996-01-01},

journal = {Review of scientific instruments},

volume = {67},

number = {10},

pages = {3757--3758},

publisher = {American Institute of Physics},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
 Kudrolli, A;  Sridhar, S
Comment on “Gaussian Orthogonal Ensemble Statistics in a Microwave Stadium Billiard with Chaotic Dynamics: Porter-Thomas Distribution and Algebraic Decay of Time Correlations” Journal Article 
In: Physical review letters, vol. 76, no. 16, pp. 3036, 1996.
Abstract | Tags: Quantum Chaos
@article{kudrolli1996comment,

title = {Comment on “Gaussian Orthogonal Ensemble Statistics in a Microwave Stadium Billiard with Chaotic Dynamics: Porter-Thomas Distribution and Algebraic Decay of Time Correlations”},

author = {A Kudrolli and S Sridhar},

year  = {1996},

date = {1996-01-01},

journal = {Physical review letters},

volume = {76},

number = {16},

pages = {3036},

publisher = {American Physical Society},

abstract = {A Comment on the Letter by H. Alt et al., Phys. Rev. Lett. 74, 62 (1995).



Received 2 March 1995

DOI:https://doi.org/10.1103/PhysRevLett.76.3036



©1996 American Physical Society},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
A Comment on the Letter by H. Alt et al., Phys. Rev. Lett. 74, 62 (1995).



Received 2 March 1995

DOI:https://doi.org/10.1103/PhysRevLett.76.3036



©1996 American Physical Society
Close
 Kudrolli, A;  Kidambi, V;  Sridhar, S
Experimental studies of chaos and localization in quantum wave functions Journal Article 
In: Physical review letters, vol. 75, no. 5, pp. 822, 1995.
Abstract | Tags: Quantum Chaos
@article{kudrolli1995experimental,

title = {Experimental studies of chaos and localization in quantum wave functions},

author = {A Kudrolli and V Kidambi and S Sridhar},

year  = {1995},

date = {1995-01-01},

journal = {Physical review letters},

volume = {75},

number = {5},

pages = {822},

publisher = {APS},

abstract = {Wave functions in chaotic and disordered quantum billiards are studied experimentally using thin microwave cavities. The chaotic wave functions display universal density distributions and density autocorrelations in agreement with expressions derived from a 0D nonlinear σ model of supersymmetry, which coincides with random matrix theory. In contrast, disordered wave functions show deviations from this universal behavior due to Anderson localization. A systematic behavior of the distribution function is studied as a function of the localization length, and can be understood in the framework of a 1D version of the nonlinear σ model.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
Wave functions in chaotic and disordered quantum billiards are studied experimentally using thin microwave cavities. The chaotic wave functions display universal density distributions and density autocorrelations in agreement with expressions derived from a 0D nonlinear σ model of supersymmetry, which coincides with random matrix theory. In contrast, disordered wave functions show deviations from this universal behavior due to Anderson localization. A systematic behavior of the distribution function is studied as a function of the localization length, and can be understood in the framework of a 1D version of the nonlinear σ model.
Close
 Prigodin, VN;  Taniguchi, Nobuhiko;  Kudrolli, A;  Kidambi, V;  Sridhar, S
Spatial correlation in quantum chaotic systems with time-reversal symmetry: Theory and experiment Journal Article 
In: Physical review letters, vol. 75, no. 12, pp. 2392, 1995.
Abstract | Tags: Quantum Chaos
@article{prigodin1995spatial,

title = {Spatial correlation in quantum chaotic systems with time-reversal symmetry: Theory and experiment},

author = {VN Prigodin and Nobuhiko Taniguchi and A Kudrolli and V Kidambi and S Sridhar},

year  = {1995},

date = {1995-01-01},

journal = {Physical review letters},

volume = {75},

number = {12},

pages = {2392},

publisher = {American Physical Society},

abstract = {The correlation between the values of wave functions at two different spatial points is examined for chaotic systems with time-reversal symmetry. Employing a supermatrix method, we find that there exist long-range Friedel oscillations of the wave function density for a given eigenstate, although the background wave function density fluctuates strongly. We show that for large fluctuations, once the value of the wave function at one point is known, its spatial dependence becomes highly predictable for increasingly large space around this point. These results are compared with the experimental wave functions obtained from billiard-shaped microwave cavities, and very good agreement is demonstrated.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
The correlation between the values of wave functions at two different spatial points is examined for chaotic systems with time-reversal symmetry. Employing a supermatrix method, we find that there exist long-range Friedel oscillations of the wave function density for a given eigenstate, although the background wave function density fluctuates strongly. We show that for large fluctuations, once the value of the wave function at one point is known, its spatial dependence becomes highly predictable for increasingly large space around this point. These results are compared with the experimental wave functions obtained from billiard-shaped microwave cavities, and very good agreement is demonstrated.
Close
 Kudrolli, A;  Sridhar, S
QUANTUM CHAOS EXPERIMENTS USING MICROWAVE CAVITIES Proceedings Article 
In: Proceedings Of The 2nd Experimental Chaos Conference, pp. 184, World Scientific 1995.
Abstract | Tags: Quantum Chaos
@inproceedings{kudrolli1995quantum,

title = {QUANTUM CHAOS EXPERIMENTS USING MICROWAVE CAVITIES},

author = {A Kudrolli and S Sridhar},

year  = {1995},

date = {1995-01-01},

booktitle = {Proceedings Of The 2nd Experimental Chaos Conference},

pages = {184},

organization = {World Scientific},

abstract = {We describe experiments using thin microwave cavities to test issues in Quantum Chaos. The experiments exploit the correspondence of the scalar Helmholtz-Maxwell and the time-independent Schrodinger equations. The eigenvalues and eigenfunctions of cavities with cross-sections in the form of chaotic billiards are obtained experimentally. In certain geometries the eigenvalue statistics are in complete agreement with Random Matrix theory. Scars are visible in some eigenfunctions, although a general rule for their obseiVation is not yet available.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {inproceedings}

}

Close
We describe experiments using thin microwave cavities to test issues in Quantum Chaos. The experiments exploit the correspondence of the scalar Helmholtz-Maxwell and the time-independent Schrodinger equations. The eigenvalues and eigenfunctions of cavities with cross-sections in the form of chaotic billiards are obtained experimentally. In certain geometries the eigenvalue statistics are in complete agreement with Random Matrix theory. Scars are visible in some eigenfunctions, although a general rule for their obseiVation is not yet available.
Close
 Sridhar, S
Cavities of Chaos Journal Article 
In: SCIENCE NEWS-WASHINGTON-, vol. 147, pp. 264–264, 1995.
Tags: Quantum Chaos
@article{sridhar1995cavities,

title = {Cavities of Chaos},

author = {S Sridhar},

year  = {1995},

date = {1995-01-01},

journal = {SCIENCE NEWS-WASHINGTON-},

volume = {147},

pages = {264--264},

publisher = {SCIENCE NEWS},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
 Kudrolli, A;  Sridhar, S;  Pandey, Akhilesh;  Ramaswamy, Ramakrishna
Signatures of chaos in quantum billiards: Microwave experiments Journal Article 
In: Physical Review E, vol. 49, no. 1, pp. R11, 1994.
Abstract | Tags: Quantum Chaos
@article{kudrolli1994signatures,

title = {Signatures of chaos in quantum billiards: Microwave experiments},

author = {A Kudrolli and S Sridhar and Akhilesh Pandey and Ramakrishna Ramaswamy},

year  = {1994},

date = {1994-01-01},

journal = {Physical Review E},

volume = {49},

number = {1},

pages = {R11},

publisher = {APS},

abstract = {The signatures of classical chaos and the role of periodic orbits in the wave-mechanical eigenvalue spectra of two-dimensional billiards are studied experimentally in microwave cavities. The survival probability for all the chaotic cavity data shows a ‘‘correlation hole,’’in agreement with theory, that is absent for the integrable cavity. The spectral rigidity Δ 3 (L), which is a measure of long-range correlation, is shown to be particularly sensitive to the presence of marginally stable periodic orbits. Agreement with random-matrix theory is achieved only after excluding such orbits, which we do by constructing a special geometry, the Sinai stadium. Pseudointegrable geometries are also studied, and are found to display intermediate behavior.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
The signatures of classical chaos and the role of periodic orbits in the wave-mechanical eigenvalue spectra of two-dimensional billiards are studied experimentally in microwave cavities. The survival probability for all the chaotic cavity data shows a ‘‘correlation hole,’’in agreement with theory, that is absent for the integrable cavity. The spectral rigidity Δ 3 (L), which is a measure of long-range correlation, is shown to be particularly sensitive to the presence of marginally stable periodic orbits. Agreement with random-matrix theory is achieved only after excluding such orbits, which we do by constructing a special geometry, the Sinai stadium. Pseudointegrable geometries are also studied, and are found to display intermediate behavior.
Close
 Sridhar, S
Quantum Dynamics of Chaotic Systems Journal Article 
In: 1993.
Tags: Quantum Chaos
@article{sridhar1993quantum,

title = {Quantum Dynamics of Chaotic Systems},

author = {S Sridhar},

year  = {1993},

date = {1993-01-01},

publisher = {Gordon and Breach, Amsterdam},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
 Sridhar, S;  Hogenboom, D;  Kudrolli, A
Exper-imental eigenvalue spectra of “rough” and multiplyconnected billiards,” Journal Article 
In: Quantum Dynamics of Chaotic Systems, eds. Yuan, JM, Feng, DH & Zaslavsky, GM (Gordon and Breach, Amsterdam), pp. 297–304, 1993.
Abstract | Tags: Quantum Chaos
@article{sridhar1993exper,

title = {Exper-imental eigenvalue spectra of “rough” and multiplyconnected billiards,”},

author = {S Sridhar and D Hogenboom and A Kudrolli},

year  = {1993},

date = {1993-01-01},

journal = {Quantum Dynamics of Chaotic Systems, eds. Yuan, JM, Feng, DH & Zaslavsky, GM (Gordon and Breach, Amsterdam)},

pages = {297--304},

abstract = {We use microwave cavities to study the spectra of billiards with" rough" perimeters, or which ara multiply connected-While simple boundaries show cumulative level densities N (E) which are in good agreement with the Weyl formula, qualitative departures are observed at low energies-when the perimeter is" roughened" by increasing the ratio of perimeter L to area A, and when obstacles are introduced to make the billiard multiply connected. Thus the Weyl formula even including perimeter corrections is not a good yardstick at low energies which probe internal length scales, interestingly, in the case of a multiply-connected billiard with a periodic array of obstacles, large" gaps" are observed in the discrete spectrum, somewhat like a solid-state lattice of finite size. The gaps persist when disorder¿ î introduced, Le. in the analog of a solid-state glass.

The influence of the shape of a planar domain on the eigenvalue …},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We use microwave cavities to study the spectra of billiards with" rough" perimeters, or which ara multiply connected-While simple boundaries show cumulative level densities N (E) which are in good agreement with the Weyl formula, qualitative departures are observed at low energies-when the perimeter is" roughened" by increasing the ratio of perimeter L to area A, and when obstacles are introduced to make the billiard multiply connected. Thus the Weyl formula even including perimeter corrections is not a good yardstick at low energies which probe internal length scales, interestingly, in the case of a multiply-connected billiard with a periodic array of obstacles, large" gaps" are observed in the discrete spectrum, somewhat like a solid-state lattice of finite size. The gaps persist when disorder¿ î introduced, Le. in the analog of a solid-state glass.

The influence of the shape of a planar domain on the eigenvalue …
Close
 Sridhar, S;  Heller, EJ
Physical and numerical experiments on the wave mechanics of classically chaotic systems Journal Article 
In: Physical Review A, vol. 46, no. 4, pp. R1728, 1992.
Abstract | Tags: Quantum Chaos
@article{sridhar1992physical,

title = {Physical and numerical experiments on the wave mechanics of classically chaotic systems},

author = {S Sridhar and EJ Heller},

year  = {1992},

date = {1992-01-01},

journal = {Physical Review A},

volume = {46},

number = {4},

pages = {R1728},

publisher = {American Physical Society},

abstract = {We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even-and odd-parity pair whose linear combination gives the localized state.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even-and odd-parity pair whose linear combination gives the localized state.
Close
 Sridhar, S;  Hogenboom, DO;  Willemsen, Balam A
Microwave experiments on chaotic billiards Journal Article 
In: Journal of statistical physics, vol. 68, no. 1-2, pp. 239–258, 1992.
Abstract | Tags: Quantum Chaos
@article{sridhar1992microwave,

title = {Microwave experiments on chaotic billiards},

author = {S Sridhar and DO Hogenboom and Balam A Willemsen},

year  = {1992},

date = {1992-01-01},

journal = {Journal of statistical physics},

volume = {68},

number = {1-2},

pages = {239--258},

publisher = {Kluwer Academic Publishers-Plenum Publishers},

abstract = {We describe experiments using billiard-shaped microwave cavities, to test ideas in quantum chaos. The experimental method for observing cavity resonances to obtain the eigenvalues, and the advantages and limitations of the techniques, including the influence of absorption, are discussed. An experimental technique to obtain a 2D mapping of the wavefunction is described. Results are displayed for 36 of the low-lying wavefunctions of a Sinai billiard cavity consisting of a central disc in a rectangular enclosure. The wavefunctions demonstrate the influence of classical periodic orbits (PO), of which there are two types: non-isolated PO, which avoid the central disc, and isolated PO, which hit the central disc. Scarred states, including those associated with isolated PO, are directly observed.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
We describe experiments using billiard-shaped microwave cavities, to test ideas in quantum chaos. The experimental method for observing cavity resonances to obtain the eigenvalues, and the advantages and limitations of the techniques, including the influence of absorption, are discussed. An experimental technique to obtain a 2D mapping of the wavefunction is described. Results are displayed for 36 of the low-lying wavefunctions of a Sinai billiard cavity consisting of a central disc in a rectangular enclosure. The wavefunctions demonstrate the influence of classical periodic orbits (PO), of which there are two types: non-isolated PO, which avoid the central disc, and isolated PO, which hit the central disc. Scarred states, including those associated with isolated PO, are directly observed.
Close
 Sridhar, S;  Parimi, PV;  Lu, WT;  Vodo, P;  Derov, John S;  Hanscom, AFRL;  Bedford, MA
Negative Refraction and Imaging in Photonic Crystals Journal Article 
In: 0000.
Abstract | Tags: Quantum Chaos
@article{sridharnegative,

title = {Negative Refraction and Imaging in Photonic Crystals},

author = {S Sridhar and PV Parimi and WT Lu and P Vodo and John S Derov and AFRL Hanscom and MA Bedford},

abstract = {Negative refraction and left-handed electromagnetism in a photonic crystal are demonstrated in waveguide and free space experiments at microwave frequencies. Precision control to achieve tailor-made refractive indices has been achieved. The negative refraction in these photonic crystals is shown to lead to imaging by a flat lens. We have also developed a generalized theory of flat lens imaging. These results promise potential applications in a variety of optical and microwave systems for communications and imaging.},

keywords = {Quantum Chaos},

pubstate = {published},

tppubtype = {article}

}

Close
Negative refraction and left-handed electromagnetism in a photonic crystal are demonstrated in waveguide and free space experiments at microwave frequencies. Precision control to achieve tailor-made refractive indices has been achieved. The negative refraction in these photonic crystals is shown to lead to imaging by a flat lens. We have also developed a generalized theory of flat lens imaging. These results promise potential applications in a variety of optical and microwave systems for communications and imaging.
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