Quantum Ergodicity Research Articles

On Quantum Unique Ergodicity for Locally Symmetric Spaces

We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally symmetric spaces to Cartan-invariant measures on an appropriate bundle. The construction uses elementary features of the representation theory of semisimple real Lie groups, and can be considered a generalization of Zelditch’s results from the upper half-plane to all locally symmetric spaces of noncompact type. This will be applied in a sequel to settle a version of the quantum unique ergodicity problem on certain locally symmetric spaces.

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.

Patterson–Sullivan Distributions and Quantum Ergodicity

We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j}, \phi_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a$ (where the sum runs over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j \to \infty$. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.

Nondiffusive phase spreading of a Bose-Einstein condensate at finite temperature

We show that the phase of a condensate in a finite temperature gas spreads linearly in time at long times rather than in a diffusive way. This result is supported by classical field simulations, and analytical calculations which are generalized to the quantum case under the assumption of quantum ergodicity in the system. This superdiffusive behavior is intimately related to conservation of energy during the free evolution of the system and to fluctuations of energy in the prepared initial state.

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical regime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".

Entropy and algorithmic complexity in quantum information theory

A theorem of Brudno says that the entropy production of classical ergodic information sources equals the algorithmic complexity per symbol of almost every sequence emitted by such sources. The recent advances in the theory and technology of quantum information raise the question whether a same relation may hold for ergodic quantum sources. In this paper, we discuss a quantum generalization of Brudno's result which connects the von Neumann entropy rate and a recently proposed quantum algorithmic complexity.

Upper Bounds on the Rate of Quantum Ergodicity

We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order $\abs{\ln\hbar}^{-1}$ on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch. We then extend the results to some classes of quantised maps on the torus and obtain a logarithmic rate for perturbed cat-maps and a sharp algebraic rate for parabolic maps.

High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators, and therefore the system remains noncommutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.

On Quantum Ergodicity for Vector Bundles

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of Hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construction of Silberman and Venkatesh [7] of representation theoretic lifts to vector bundles.

Quantum Unique Ergodicity for Maps on the Torus

When a map is classically uniquely ergodic, it is expected that its quantization will posses quantum unique ergodicity. In this paper we give examples of Quantum Unique Ergodicity for the perturbed Kronecker map, and an upper bound for the rate of convergence.

Quantum Variance and Ergodicity for the Baker's Map

We prove a Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak is that every eigenfunction ϕn of the Laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En ∞); that is, “strong scars” are absent. We study numerically the rate of equidistribution for a uniformly hyperbolic, Sinai-type, planar Euclidean billiard with Dirichlet boundary condition (the “drum problem”) at unprecedented high E and statistical accuracy, via the matrix elements 〈ϕn, Âϕm〉 of a piecewise-constant test function A. By collecting 30,000 diagonal elements (up to level n ≈ 7 × 105) we find that their variance decays with eigenvalue as a power 0.48 ± 0.01, close to the semiclassical estimate ½ of Feingold and Peres. This contrasts with the results of existing studies, which have been limited to En a factor 102 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance as a function of distance from the diagonal, against Feingold-Peres (or spectral measure) at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions. © 2006 Wiley Periodicals, Inc.

Invariant measures and arithmetic quantum unique ergodicity

We classify measures on the locally homogeneous space ?i SL(2,R) ?~ L which are invariant and have positive entropy under the diagonal subgroup of SL(2,R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result.

On the scarring of eigenstates in some arithmetic hyperbolic manifolds

We deal here with the conjecture of Quantum Unique Ergodicity in a specific case. Rudnick and Sarnak showed in [Rudnick Z. and Sarnak P.: The Behaviour of Eigenstates of Arithmetic Hyperbolic Manifolds. Communications in Mathematical Physics 161 (1994), 195–213] that there is no strong scarring on closed geodesics for compact arithmetic congruence surfaces derived from quaternion division algebras (see theorem 1.1). We extend this theorem to a class ( K 2 S ) of Riemannian manifolds derived from quaternion division algebras, and show that there is no strong scarring on closed geodesics or on Γ R -closed imbedded totally geodesic surfaces in these manifolds (see theorem 4.1).

Proof of the Kurlberg–Rudnick rate conjecture

Proof of the Kurlberg–Rudnick rate conjecture

Quantum Leaks

We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.

Ergodic Quantum Computing

We propose a (theoretical) model for quantum computation where the result can be read out from the time average of the Hamiltonian dynamics of a 2-dimensional crystal on a cylinder.The Hamiltonian is a spatially local interaction among Wigner–Seitz cells containing six qubits. The quantum circuit that is simulated is specified by the initialization of program qubits. As in Margolus’ Hamiltonian cellular automaton (implementing classical circuits), a propagating wave in a clock register controls asynchronously the application of the gates. However, in our approach all required initializations are basis states. After a while the synchronizing wave is essentially spread around the whole crystal. The circuit is designed such that the result is available with probability about 1/4 despite of the completely undefined computation step. This model reduces quantum computing to preparing basis states for some qubits, waiting, and measuring in the computational basis. Even though it may be unlikely to find our specific Hamiltonian in real solids, it is possible that also more natural interactions allow ergodic quantum computing.

Microscopic Theory for the Quantum to Classical Crossover in Chaotic Transport

We present a semiclassical theory for the scattering matrix S of a chaotic ballistic cavity at finite Ehrenfest time. Using a phase-space representation coupled with a multibounce expansion, we show how the Liouville conservation of phase-space volume decomposes S as S=S(cl) plus sign in circle S(qm). The short-time, classical contribution S(cl) generates deterministic transmission eigenvalues T=0 or 1, while quantum ergodicity is recovered within the subspace corresponding to the long-time, stochastic contribution S(qm). This provides a microscopic foundation for the two-phase fluid model, in which the cavity acts like a classical and a quantum cavity in parallel, and explains recent numerical data showing the breakdown of universality in quantum chaotic transport in the deep semiclassical limit. We show that the Fano factor of the shot-noise power vanishes in this limit, while weak localization remains universal.

The Data Compression Theorem for Ergodic Quantum Information Sources

We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate.