aa r X i v : . [ qu a n t - ph ] M a r Quantum multiresolution: tower of scales
Antonina N. Fedorova, Michael G. Zeitlin [email protected], [email protected]
Abstract.
We demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic,entangled or decoherent, from the basic localized modes in various collective models arisingfrom the quantum hierarchy described by Wigner-like equations. The numerical simulationdemonstrates the formation of various (meta) stable patterns or orbits generated by internalhidden symmetry from generic high-localized fundamental modes. In addition, we can controlthe type of behaviour on the pure algebraic level by means of properly reduced algebraic systems(generalized dispersion relations).
1. Introduction. New localized modes and patterns: why need we them?
It is widely known that the currently available experimental techniques in the area of quantumphysics as well as the present level of the understanding of phenomenological models, outstripsthe actual level of mathematical description. Considering the problem of describing the reallyexisting and/or realizable states, one should not expect that (gaussian) coherent states would beenough to characterize complex quantum phenomena. The complexity of a set of relevant states,including entangled (chaotic) ones is still far from being clearly understood and moreover frombeing realizable [1]. Our motivations arise from the following general questions [2]: how can werepresent a well localized and reasonable state in mathematically correct form? is it possible tocreate entangled and other relevant states by means of these new localized building blocks? Thegeneral idea is rather simple: it is well known that the generating symmetry is the key ingredientof any modern reasonable physical theory. Roughly speaking, the representation theory of theunderlying (internal/hidden) symmetry (classical or quantum, finite or infinite dimensional,continuous or discrete) is the useful instrument for the description of (orbital) dynamics. Theproper representation theory is well known as “local nonlinear harmonic analysis”, in particularcase of the simple underlying symmetry, affine group, aka wavelet analysis. From our point ofview the advantages of such approach are as follows: i) the natural realization of localized statesin any proper functional realization of (Hilbert) space of states, ii) the hidden symmetry of achosen realization of the functional model describes the (whole) spectrum of possible states viathe so-called multiresolution decomposition. Effects we are interested in are as follows: ahierarchy of internal/hidden scales (time, space, phase space); non-perturbative multiscales:from slow to fast contributions, from the coarser to the finer level of resolution/decomposition; .the coexistence of the levels of hierarchy of multiscale dynamics with transitions betweenscales; the realization of the key features of the complex quantum world such as the existencef chaotic and/or entangled states with possible destruction in “open/dissipative” regimes dueto interactions with quantum/classical environment and transition to decoherent states.N-particle Wigner functions allow to consider them as some quasiprobabilities. The fulldescription for quantum ensemble can be done by the hierarchy of functions (symbols): W = { W s ( x , . . . , x s ) , s = 0 , , . . . } which are solutions of Wigner equations: ∂W n ∂t = − pm ∂W n ∂q + ∞ X ℓ =0 ( − ℓ (¯ h/ ℓ (2 ℓ + 1)! ∂ ℓ +1 U n ( q ) ∂q ℓ +1 ∂ ℓ +1 W n ∂p ℓ +1 . (1)The similar equations describe the important decoherence processes.
2. Variational multiresolution representation
We obtain our multiscale/multiresolution representations for solutions of Wigner-like equations(1) via the variational-wavelet approach [2] and represent the solutions as decomposition intolocalized eigenmodes related to the hidden underlying set of scales: W n ( t, q, p ) = ∞ M i = i c W in ( t, q, p ) , where value i c corresponds to the coarsest level of resolution c in the full multiresolutiondecomposition (MRA) [3] of the underlying functional space: V c ⊂ V c +1 ⊂ V c +2 ⊂ . . . and p = ( p , p , . . . ) , q = ( q , q , . . . ) , x i = ( p , q , . . . , p i , q i ) are coordinates in phase space.We introduce the Fock-like space structure on the whole space of internal hidden scales H = M i O n H ni for the set of n-partial Wigner functions (states): W i = { W i , W i ( x ; t ) , . . . , W iN ( x , . . . , x N ; t ) , . . . } , where W p ( x , . . . , x p ; t ) ∈ H p , H = C, H p = L ( R p ) (or any different proper functionalspace), with the natural Fock space like norm:( W, W ) = W + X i Z W i ( x , . . . , x i ; t ) i Y ℓ =1 µ ℓ . First of all, we consider W = W ( t ) as a function of time only, W ∈ L ( R ), via multiresolutiondecomposition which naturally and efficiently introduces an infinite sequence of the underlyinghidden scales. We have the contribution to the final result from each scale of resolution fromthe whole infinite scale of spaces. The closed subspace V j ( j ∈ Z ) corresponds to the level j ofresolution and satisfies the following properties: let D j be the orthonormal complement of V j with respect to V j +1 : V j +1 = V j L D j . Then we have the following decomposition: { W ( t ) } = M −∞ 3. Conclusions By using proper high-localized bases on orbits generated by actions of internal hidden symmetriesof underlying functional spaces, we can describe and classify the full zoo of patterns with non-trivial behaviour including localized (coherent) structures in quantum systems with complicatedbehaviour (Figs. 1, 2). The numerical simulation demonstrates the formation of various (meta)stable patterns or orbits generated by internal hidden symmetry from generic high-localizedfundamental modes. These (nonlinear) eigenmodes are more realistic for the modeling ofclassical/quantum dynamical process than the (linear) gaussian-like coherent states. Here wemention only the best convergence properties of the expansions based on wavelet packets, whichrealize the minimal Shannon entropy property and the exponential control of the convergence ofexpansions like (3). Figs. 1, 2 demonstrate the steps of (hidden) multiscale resolution, startingfrom coarse–graining, during the full quantum interaction/evolution of entangled states leadingto the growth of the degree of complexity (entanglement) of the quantum state. It should benoted that we can control the type of behaviour on the level of the reduced algebraic system(Generalized Dispersion Relation) (2) [2]. Figure 1. Entangled Wigner function. W ( q , p ) Figure 2. Localized (decoherent) pattern:(waveleton) Wigner function. References [1] D. Sternheimer, Deformation Quantization: Twenty Years After, math/9809056; W. P. Schleich, QuantumOptics in Phase Space (Wiley, 2000); W. Zurek, Decoherence, einselection, and the quantum origins of theclassical, quant-ph/0105127.[2] http://math.ipme.ru/zeitlin.html or http://mp.ipme.ru/zeitlin.html[3] Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, 1990); F. Auger e.a., Time-Frequency Toolbox (CNRS, 1996); D. Donoho,