Quantization Of Manifolds Research Articles

Proposal of ensemble qubits with two-atom decay

We propose and analyze a novel approach to implement ensemble qubits. The required anharmonicity is provided by a simultaneous decay of two atoms (i.e. two-atom decay), which is achieved by fully quantum degenerate parametric amplification. For an atomic ensemble, the two-atom decay generates and stabilizes a 2D quantum manifold, which is spanned by the ground and single-excited superradiant states. Moreover, this nonlinear decay process can strongly suppress transitions to higher-excited superradiant states, and convert residual transitions into an effective decay from the single-excitation superradiant state to the ground state. Our method does not require Rydberg dipole blockade and, thus, strong atom-atom interactions, compared to previous work. This indicates that it can be applied to typical atomic or spin ensembles in simple experimental setups. Remarkably, our idea is compatible with the cavity protection mechanism, and therefore spin dephasing due to inhomogeneous broadening can be strongly suppressed. The presented ensemble qubit provides a new platform for quantum information processing, and also extends the range of applications of atomic or spin ensembles.

Discrete quantum geometry and intrinsic spin Hall effect

We show that the quantum geometry of the Fermi surface can be numerically described by a 3-dimensional discrete quantum manifold. This approach not only avoids singularities in the Fermi sea, but it also enables the precise computation of the intrinsic Hall conductivity resolved in spin, as well as any other local properties of the Fermi surface. The method assures numerical accuracy when the Fermi level is arbitrarily close to singularities, and it remains robust when Kramers degeneracy is protected by symmetry. The approach is demonstrated by calculating the anomalous Hall and spin Hall conductivities of a 2-band lattice model of a Weyl semimetal and a full-band ab-initio model of zincblende GaAs.

Correspondence between light-absorption spectrum and nonequilibrium work distribution as a mean to access free energy differences between electronic states.

The problem of recovering the free energy difference between two electronic states has been investigated by Frezzato [Chem. Phys. Lett. 533, 106 (2012)], exploring the equivalence between light-absorption spectra and work distribution, hence opening to the application of a spectroscopic version of the Jarzynski equality (JE) [Phys. Rev. Lett. 78, 2690 (1997)]. Here, assuming the validity of the time-dependent perturbation theory, we demonstrate that such equivalence does not lead to the known form of the JE. This is ascribed to the fact that light-absorption processes cannot be described as stochastic processes. To emphasize such an aspect, we devise a stochastic model for the UV-vis (ultraviolet and visible) absorption, suitable for determining the free energy difference between two generic quantum manifolds in a JE-like fashion. However, the model would require explicit knowledge of the transition dipole moments, which are in general not available. Nonetheless, we derive a spectroscopic version of the JE that allows us to recover the free energy difference between the ground and an excited electronic state when the latter state is the only one observed in the spectrum.

Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation

New experiments present experimental evidence for the dynamical quantum Zeno effect, in which a stabilizing friction leads to a continuous phase change among degenerate states in a quantum superposition.

Generating higher-order quantum dissipation from lower-order parametric processes

The stabilisation of quantum manifolds is at the heart of error-protected quantum information storage and manipulation. Nonlinear driven-dissipative processes achieve such stabilisation in a hardware efficient manner. Josephson circuits with parametric pump drives implement these nonlinear interactions. In this article, we propose a scheme to engineer a four-photon drive and dissipation on a harmonic oscillator by cascading experimentally demonstrated two-photon processes. This would stabilise a four-dimensional degenerate manifold in a superconducting resonator. We analyse the performance of the scheme using numerical simulations of a realisable system with experimentally achievable parameters.

Quantum engineering. Confining the state of light to a quantum manifold by engineered two-photon loss.

Physical systems usually exhibit quantum behavior, such as superpositions and entanglement, only when they are sufficiently decoupled from a lossy environment. Paradoxically, a specially engineered interaction with the environment can become a resource for the generation and protection of quantum states. This notion can be generalized to the confinement of a system into a manifold of quantum states, consisting of all coherent superpositions of multiple stable steady states. We have confined the state of a superconducting resonator to the quantum manifold spanned by two coherent states of opposite phases and have observed a Schrödinger cat state spontaneously squeeze out of vacuum before decaying into a classical mixture. This experiment points toward robustly encoding quantum information in multidimensional steady-state manifolds.

Excited state electronic landscape of mPlum revealed by two-dimensional double quantum coherence spectroscopy.

Red fluorescent proteins (RFPs) are widely used probes for monitoring subcellular processes with extremely high spatial and temporal precision. In this work, we employed spectrally resolved transient absorption (SRTA) and two-dimensional double quantum coherence (2D2Q) spectroscopy to investigate the excited state electronic structure of mPlum, a well-known RFP. The SRTA spectra reveal the presence of excited state absorption features at both the low- and high-energy sides of the dominant ground state bleach contribution. The 2D2Q spectra measured at several excitation wavelengths reveal a peak pattern consistent with the presence of more than three electronic states (i.e., ground, excited, and doubly excited). Numerical modeling of this response suggests that the features are consistent with a 1-1-2 electronic structure. The two closely spaced (∼1500 cm(-1)) levels in the double quantum manifold appear at opposite anharmonicities relative to twice the energy of the lowest energy transition. These observations explain the excited state absorption contributions observed in spectrally resolved transient grating and transient absorption measurements and demonstrate the utility of multidimensional spectroscopy in unraveling congested spectra relative to conventional one-dimensional methods.

On the Geometry and Quantization of Manifolds of Positive Semi-Definite Matrices

The geometry of different spaces of positive semi-definite matrices buffeted by rank and trace constraints is studied. In addition to revealing their Riemannian structure, we derive the normalized volume of a ball over these spaces. Further, we use the leading coefficient from the ball volume expansion to bound the quantization error incurred with finite-sized sphere-packing codebooks as well as random codebooks to represent sources distributed over general Riemannian manifolds.

A Rindler-KAM Spacetime Geometry and Scaling the Planck Scale Solves Quantum Relativity and Explains Dark Energy

We introduce an ultra high energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive an effective quantum gravity energy-mass relation and compute a dark energy density in complete agreement with all cosmological measurements, specifically WMAP and type 1a supernova. In particular we find that ordinary measurable energy density is given by E1= mc2 /22 while the dark energy density of the vacuum is given by E2 = mc2 (21/22). The sum of both energies is equal to Einstein’s energy E = mc2. We conclude that E= mc2 makes no distinction between ordinary energy and dark energy. More generally we conclude that the geometry and topology of quantum entanglement create our classical spacetime and glue it together and conversely quantum entanglement is the logical consequence of KAM theorem and zero measure topology of quantum spacetime. Furthermore we show via our version of a Rindler hyperbolic spacetime that Hawking negative vacuum energy, Unruh temperature and dark energy are different sides of the same medal.

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

An $f$-structure on a manifold $M$ is an endomorphism field $\phi\in\Gamma(M,\End(TM))$ such that $\phi^3+\phi=0$. Any $f$-structure $\phi$ determines an almost CR structure $E_{1,0}\subset T_\C M$ given by the $+i$-eigenbundle of $\phi$. Using a compatible metric $g$ and connection $\nabla$ on $M$, we construct an odd first-order differential operator $D$, acting on sections of $\S=\Lambda E_{0,1}^*$, whose principal symbol is of the type considered in arXiv:0810.0338. In the special case of a CR-integrable almost $\S$-structure, we show that when $\nabla$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator $D$ is given by $D = \sqrt{2}(\dbbar+\dbbar^*)$, where $\dbbar$ is the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with $f$-structure that reduce to familiar methods in symplectic geometry in the case that $\phi$ is a compatible almost complex structure, and to the contact quantization defined in \cite{F4} when $\phi$ comes from a contact metric structure. The first is an index-theoretic approach involving the operator $D$; for certain group actions $D$ will be transversally elliptic, and using the results in arXiv:0810.0338, we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization.

Geometric Quantization in the Non-compact Setting

The purpose of the workshop was to bring together mathematicians interested in ”quantization of manifolds” in a broad sense: given classical data, such as a Lie group G acting on a symplectic manifold M , construct a quantum version, that is a representation of G in a vector space Q ( M ) reflecting the classical properties of M .

Hausdorff dimension of a particle path in a quantum manifold

After recalling the concept of the Hausdorff dimension, we study the fractal properties of a quantum particle path. As a novelty we consider the possibility for the space where the particle propagates to be endowed with a quantum-gravity-induced minimal length. We show that the Hausdorff dimension accounts for both the quantum mechanics uncertainty and manifold fluctuations. In addition the presence of a minimal length breaks the self-similarity property of the erratic path of the quantum particle. Finally we establish a universal property of the Hausdorff dimension as well as the spectral dimension: They both depend on the amount of resolution loss which affects both the path and the manifold when quantum gravity fluctuations occur.

Spectral dimension of a quantum universe

In this paper, we calculate in a transparent way the spectral dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a spectral dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the spectral dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.

Surgery and bordism groups in quantum partial differential equations. I: The quantum Poincaré conjecture

In this work, in two parts, we continue to develop the geometric theory of quantum PDE’s, introduced by us starting from 1996. (The second part is quoted in Prástaro [A. Prástaro, Surgery and bordism groups in quantum partial differential equations. II: Variational quantum PDE’s, Nonlinear Anal. TMA, in press ( 10.1016/j.na.2008.10.063)]) This theory has the purpose to build a rigorous mathematical theory of PDE’s in the category D S of noncommutative manifolds ( quantum (super)manifolds), necessary to encode physical phenomena at microscopic level (i.e., quantum level). Aim of the present paper is to report on some new issues in this direction, emphasizing an interplaying between surgery, integral bordism groups and conservations laws. In particular, a proof of the Poincaré conjecture, generalized to the category D S , is given by using our geometric theory of PDE’s just in such a category.

Surgery and bordism groups in quantum partial differential equations, II: Variational quantum PDEs

This is the second part of a work devoted to the interplay between surgery, integral bordism groups and conservation laws, in order to characterize the geometry of PDEs in the category Q S of quantum (super)manifolds. (Part I is quoted in Ref. [A. Prástaro, Surgery and bordism groups in quantum partial differential equations, I: The quantum Poincaré conjecture, Nonlinear Anal. TMA, in press ( doi:10.1016/j.na.2008.11.077)].) In this paper we will consider variational problems, in the category Q S , constrained by partial differential equations. We get theorems of existence for local and global solutions. The characterization of global solutions is made by means of integral bordism groups. Applications to some important examples of Mathematical Physics, as quantum super-black-hole solutions of quantum super-Yang–Mills equations, are discussed in some detail. Quantum supermanifolds allow us to unify, at the quantum level, the four fundamental forces, (gravitational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geometric structure. The geometric theory of PDEs, built in the category Q S of quantum supermanifolds, gives us the right mathematic tool to describe quantum phenomena also at very high energy levels, where quantum gravity becomes dominant.

(Co)bordism groups in quantum super PDEs. II: Quantum super PDEs

Following our previous works on the integral (co)bordism groups of quantum PDEs [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing Co., Singapore, 1996, 760pp; A. Prástaro, (Co)bordisms in PDE's and quantum PDE's, Rep. Math. Phys. 38(3) (1996) 443–455; A. Prástaro, Quantum and integral (co)bordism groups in partial differential equations, Acta Appl. Math. 51(3) (1998) 243–302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59(2) (1999) 111–202; A. Prástaro, (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co., Singapore, 2004, 500pp; A. Prástaro, Quantum super Yang-Mills equations: global existence and mass-gap, Dynamic Syst. Appl. 4 (2004) 227–234; A. Prástaro, Th.M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sci. Paris 330(I-7) (2000) 545–550; A. Prástaro, Th.M. Rassias, Results on the J.D’Alembert equation, Ann Acad. Paed. Crac. Stud. Math. 1 (2001) 117–128.], we specialize, now, on quantum super partial differential equations, i.e., partial differential equations built in the category of quantum supermanifolds. These are manifolds modeled on locally convex topological vector spaces built starting from quantum algebras endowed also with a Z 2 -gradiation, and a Z 2 -graded Lie algebra structure, ( quantum superalgebra). Then, we extend to these manifolds, with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDEs, arising from the Dirac quantization of some classical (super) PDEs, are considered in some details.

(Co)bordism groups in quantum super PDE's. I: Quantum supermanifolds

Following our previous works on noncommutative manifolds and noncommutative PDE's [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing, Singapore, 1996, 760pp; (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; Quantized Partial Differential Equations, World Scientific Publishing, Singapore, 2004, 500pp; Quantum super Yang–Mills equations: global existence and mass-gap, Dyn. Syst. Appl. 4 (2004) 227–234], we consider in these series of three papers, some further results on quantum supermanifolds and quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal.: Real World Appl., to appear; (Co)bordism groups in quantum super PDE's. III: quantum super Yang–Mills equations, Nonlinear Anal.: Real World Appl., to appear]. In particular, in this first part we focus our attention on quantum supermanifolds. These structures globalize the notion of quantum superalgebras, obtaining noncommutative manifolds that are useful to give a fully covariant description of noncommutative geometric structures, hence of quantum physics. We study the geometry of quantum supermanifolds and characterize their (co)homological and (co)bordism properties. Covariant quantizations of super PDE's are given, obtaining examples of quantum supermanifolds justifying their definitions.

Moduli of quantum Riemannian geometries on ⩽4 points

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously [S. Majid, Commun. Math. Phys. 225, 131 (2002)]. The full moduli space is found for ⩽3 points, and a restricted moduli space for 4 points. Generalized Levi–Cività connections and their curvatures are found for a variety of models including models of a discrete torus. The topological part of the moduli space is found for ⩽9 points based on the known atlas of regular graphs. We also remark on aspects of quantum gravity in this approach.

Berry phase effects in two-level and three-level atoms

Group-theoretical methods are developed for treating Berry phase effects, which are related to Cartan subalgebra. The theory is applied to two-level and three-level atoms interacting with perturbations that are described by the SU(2) or SU(3) algebra. By using fiber-bundle theories, it is found that a time development operator that depends on Cartan group generators can represent a fiber while a time development operator that depends on other generators of the group represents the base of the quantum manifold. The total time development operator is obtained by multiplication of these two parts and the fiber-bundle theory is applied for calculating Berry phase effects. Explicit expressions for Berry phases are obtained under the adiabatic approximation.

Quantum space-times in the year 2002

We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $ CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics.