Quantum Formula Research Articles

Adiabatic driving, geometric phases, and the geometric tensor for classical states

We use the Hilbert space formulation of classical mechanics, known as the Koopman–von Neumann formalism, to study adiabatic driving, geometric phases, and the geometric tensor for classical states. In close relation to what happens to a quantum state, a classical Koopman–von Neumann eigenstate will acquire a geometric phase factor expiΦ after a closed variation of the parameters λ in its associated Hamiltonian. The explicit form of Φ is then derived for integrable systems, and its relation with the Hannay angle is shown. Additionally, we use quantum formulas to write an adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase. We also define a classical analog of the geometric tensor, thus defining a Fubini–Study metric for classical states, and we use the singularities of the tensor to link the transition from Arnold–Liouville integrability to chaos with some of the mathematical formalism of quantum phase transitions. While the formulas and definitions we use originate in quantum mechanics, all the results found are purely classical, no classical or semiclassical limit is ever taken.

Quantum–classical correspondence and dissipative to dissipationless crossover in magnetotransport phenomena

The three-dimensional magneto-conductivity tensor was derived in a gauge invariant form based on the Kubo formula considering quantum effects under a magnetic field, such as the Landau quantization and quantum oscillations. We analytically demonstrated that the quantum formula of the magneto-conductivity can be obtained by adding a quantum oscillation factor to the classical formula. This result establishes the quantum–classical correspondence, which has long been missing in magnetotransport phenomena. Moreover, we found dissipative-to-dissipationless crossover in the Hall conductivity by paying special attention to the analytic properties of the thermal Green’s function. Finally, by calculating the magnetoresistance of semimetals, we identified a phase shift in quantum oscillation originating from the dissipationless transport predominant at high fields.

A new method for detecting dark Matter and explaining formulas which describe fluid matter

Nowadays there is a new challenge among scientists, all of us know dark energy spreads the universe by constant index, but the big question is whether the universe expanding without stopping. I am not agreeing, because on the other side, we have dark matter and we know the force of dark matter and dark energy are opposite of each other. In this article, it has been tried to explain how to detect dark fluid and made an effort to find a formula that describe the relationship between dark matter and dark energy or totally dark fluid. Throughout the history, many scientists had the idea to estimate the behavior of dark matter by the means of time, it means they desired to see what is the effect of time on the movement of dark fluid and influence of it on expanding and diminishing physical universe. Cosmology, attached to the theory of dark matter and dark energy, elaborates that both of them are in a single framework and they are not separated into ptomaine. In the last section I added time to predict dark fluid treatment by passing time. To pursue this goal, at first an experiment was designed in which by assist of cigarette smoke and electrical light, the behavior of dark fluid was observed by the author. Then, by means of quantum physic equations and adding time as a stable factor to quantum physic equation, a model and formula was coined by which it was possible to show dark fluid movement. It is very significant to state that this final formula was not completely accomplished due to the fact the gravity of black hole was not regarded in this experiment. Asan important factor, by help of cosmic noise, voyager program of NASA, information theory is related to lost galaxy information.

Index Theory of Chiral Unitaries and Split-Step Quantum Walks

Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.

Transition radiation from a Dirac-particle wave packet traversing a mirror

The explicit expression for the inclusive probability to record a photon created in transition radiation from a one Dirac particle wave packet traversing an ideally conducting plate is derived in the leading order of perturbation theory. The anomalous magnetic moment of the Dirac particle is taken into account. It is shown that the quantum corrections to transition radiation from an electrically charged particle give rise to production of photons with polarization vector orthogonal to the reaction plane ($E$-plane). These corrections result from both the quantum recoil and the finite size of a wave packet. As for transition radiation produced by a neutron falling normally onto the conducting plate, the probability to detect a photon with polarization vector lying in the reaction plane does not depend on the observation angle and the energy of the incident particle. The peculiarities of transition radiation stemming from different shapes of the particle wave packet are investigated. In particular, the transition radiation produced by the wave packet of one twisted Dirac particle is described. The comparison with classical approach to transition radiation is given and the quantum formula for the inclusive probability to detect a photon radiated by the $N$-particle wave packet is derived.

Beyond the Formulations of the Fluctuation Dissipation Theorem Given by Callen and Welton (1951) and Expanded by Kubo (1966)

The quantum formula of the fluctuation dissipation theorem (FDT) was given by Callen and Welton in 1951 for the case of conductors, and then expanded by Kubo in 1966. The drawback of these quantum relations concerns with the appearance of a zero-point contribution, ${\hbar \omega}/{2}$ with $\hbar$ the reduced Planck constant and $\omega$ the angular frequency of the considered photon, which implies a divergence of the fluctuation spectrum at increasing frequencies. This divergence is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum. As a consequence, the quantum formulation of the FDT as given by Callen-Welton and Kubo introduces a Field Grand Challenge associated with the existence or less of a vacuum-fluctuations catastrophe for the energy-density spectrum. Here we propose a solution to this challenge by taking into account of the Casimir energy that, in turns, is found to be responsible for a quantum correction of the Stefan-Boltzmann law.

Quantum Extropy and Statistical Properties of the Radiation Field for Photonic Binomial and Even Binomial Distributions

We present quantum formula for extropy based on the Husimi Q-function and the atomic Husimi Q-function. The quantum extropy is used to detect the entanglement or nonlocal correlations of the system consisted of a single qubit and the radiation field. We provide explicit forms of the binomial and even binomial distributions. We compare the temporal behavior of quantum extropy within the atomic and field bases with the tomographic entropy, which is a measure for quantifying nonlocal correlations between the field and a single qubit. The photon statistics of the field can also be quantified by the evolution of the Mandel parameter, if the field initially follows the binomial and even binomial distributions. We use the total density matrix to compute and analyze the time evolution of the initial photonic binomial probability distribution that governs the behavior of the atom–photon entanglement. In this context, we investigate the links among the temporal behavior of the atomic quantum extropy, tomographic entropy, and statistical properties of the field. The quantum extropy is a useful indicator of the qubit–field entanglement, whereas the quantum extropy of the field basis is a good indicator of the dynamical behavior of the atom–field system.

Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation

In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.

Relationships between identities for quantum Bernstein bases and formulas for hypergeometric series

Two seemingly disparate mathematical entities - quantum Bernstein bases and hypergeometric series - are revealed to be intimately related. The partition of unity property for quantum Bernstein bases is shown to be equivalent to the Chu-Vandermonde formula for hypergeometric series, and the Marsden identity for quantum Bernstein bases is shown to be equivalent to the Pfaff-Saalsch?tz formula for hypergeometric series. The equivalence of the q-versions of these formulas and identities is also demonstrated.

Variance of the sum of independent quantum computing errors

The sum of quantum computing errors is the key element both for the estimation and control of errors in quantum computing and for its statistical study. In this article we analyze the sum of two independent quantum computing errors, X_1 and X_2, and we obtain the formula of the variance of the sum of these errors: V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}. We conjecture that this result holds true for general quantum computing errors and we prove the formula for independent isotropic quantum computing errors.

Relativistic mass due to a dilatant vacuum leads to a quantum reformulation of the relativistic kinetic energy

Relativistic mass change with speed is considered here as the effect of a viscous, dilatant vacuum, whose apparent viscosity is related to the Lorentz factor. Transient solid-like vacuum due to shear stress is presented as the reason why vacuum prevents the speed of massive objects from being indefinitely increased. Such a vacuum – that in a previous study allowed to exactly calculate the Pioneer anomaly, Mercury’s perihelion precession, and was shown to be compatible with stable planetary orbits – leads here to a quantum formula for the relativistic kinetic energy. A formula which distinguishes between the case of accelerated charges in a vacuum, for which a Stokes–Einstein radius comes into play, and the case of accelerated macroscopic bodies, for which the quantum potential term vanishes. In this way, incidentally, one obtains again correct results for the Pioneer 10, confirming the role of vacuum’s viscous force. This description of a quantum mechanism underlying the relativistic kinetic energy may be also helpful in constructing a theory of quantum relativity and may even tell us more about the interactions of matter with the Higgs field and the dark sector: two issues which can be themselves linked to a dilatant vacuum.

Quantum photodetachment of hydrogen negative ion in a harmonic potential subjected to static electric field

Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H− in the perturbed one dimensional linear harmonic potential via static electric field. To this end, the quantum formula for total photodetachment cross section of the H− ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schrödinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB (Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy, oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A.

Noisy frequency estimation with noisy probes

We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from n + 1 qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called ‘Zeno’ scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.

Surface roughness modulated resistivity in copper thin films

The surfaces of metallic thin films are never flat. The resistivity in thin films is very different from that in bulk because of the unavoidable rough surfaces. In this study, we apply a quantum-mechanical method to study the resistivity in metallic thin films. The resulting resistivity formula for metallic thin films merely involves two parameters: bulk relaxation time and surface roughness. We use the formula to fit a large number of experimental data sets for copper thin films obtained using different growing methods. With an additional tuning parameter for calibrating the film thickness, the quantum formula can provide a universal fitting to most data with a satisfactory precision, regardless of their growing methods or data source.

Khul’i (Right of Redemption) under Sharia Courts Rules of Bauchi State, Nigeria, 2013: Issues and Challenges

In the year 2000, most of the States in Northern Nigeria re-introduced Sharia as part of their legal system in disposing of both civil and criminal cases. Bauchi State was among the States that re-introduced Sharia as part of its legal system. Based on that, Sharia Courts were created to replace Area Courts, and Sharia Courts Civil Procedure Rules was enacted to regulate the proceedings before the said Sharia Courts on Civil Matters. The Bauchi State Sharia Courts Civil Procedure Rules, 2001 did not contain a provision on Khul’i. Prior to the amendment of Bauchi State Sharia Courts Civil Procedure Rules, Judges of Sharia Courts were in a serious dilemma as to the amount of money to be paid by a woman in order to relieve herself from the bond of marriage. This is because Khul’i is a right accorded to a wife to remove herself from the union of marriage (to get a divorce) by compensating her husband. In order to address the above problem and others, Sharia Courts Civil Procedure Rules of Bauchi State, 2001 was amended in 2013 and right of Khul’i (Redemption) was incorporated under Order 21 of the Bauchi State Civil Procedure Rules, 2013. With the amendment of the said Sharia Courts Civil Procedure Rules of Bauchi State, instead of addressing the problem of a quantum formula for Khul’i, rather brought a controversy to the said issue. Therefore, this study critically examines the Qur’anic Verses, Hadith and the opinion of the Islamic Scholars on Khul’i and compare them with Order 21 Right of Redemption (Khul’i) of Sharia Courts Civil Procedure (Amendment) Rules of Bauchi State, 2013 so as to ascertain its consistency or otherwise.

Comment on “Comparative study of ab initio nonradiative recombination rate calculations under different formalisms”

In a recent article [Phys. Rev. B 91, 205315 (2015)] Shi, Xu, and Wang presented a comparison of several formalisms to calculate nonradiative recombination rates and concluded the ``one-dimensional (1D) quantum formula'' that was used by Alkauskas et al. [Phys. Rev. B 90, 075202 (2014)] is insufficient to accurately describe nonradiative capture rates. Our analysis of the results of Shi, Xu, and Wang indicates that their conclusions about the 1D quantum formula are unfounded and stem from an error in their calculations. Our own calculations demonstrate that the 1D quantum formula approach yields reliable and accurate results for nonradiative recombination rates.

Probabilities of Bremsstrahlung Emission of Photons at Low-Energy Electron-Nuclear Collisions in a Magnetic Field

We consider quantum-mechanical probabilities of bremsstrahlung of photons in the case of low-energy Coulomb collisions in a magnetic field, where the scattering center so perturbs the state of the incident electron that the motion of the latter becomes quasi-bound. Quantum formulas for the spectral power of bremsstrahlung radiation are obtained from the classical formulas by replacing the Fourier amplitudes of the particle velocity with matrix elements of the velocity operator for wave functions, which are normalized by the condition of a unit flux being incident on the nucleus (or an equivalent outgoing flux), with summation over finite Landau levels and quantized values of the impact parameter. Equivalent forms of the specified matrix elements, which are expressed in terms of the Coulomb field and annihilation/creation operators for the eigenfunctions of the operator of the squared impact parameter, are presented. The obtained presentations for the spectral power of bremsstrahlung radiation in the case of quasi-bound electron motion allow one to translate the results of calculating this value in the classical limit to the quantum case, which is typical of white dwarfs with the strongest magnetic fields.

Quantum Behavior Arises Because Our Universe is a Fractal

To explain the origin of quantum behavior, we propose a fractal calculus to describe the non-local property of the fractal curve [Y. Tao, J. Appl. Math. 2013 (2013) 308691]. This study demonstrates that if the dimension of time axis is slightly less than 1, then Planck’s energy quantum formula will naturally emerge. In this paper, we further show that if the dimension of time axis is less than 1, Heisenberg’s Principle of Uncertainty will emerge as well. Our finding implies that fractal calculus may be an intrinsic way of describing quantum behavior. To test our theory, we also provide an experimental proposal for measuring the dimension of time axis.

Uniform Additivity in Classical and Quantum Information.

Information theory quantifies the optimal rates of resource interconversions, usually in terms of entropies. However, nonadditivity often makes evaluating entropic formulas intractable. In a few auspicious cases, additivity allows a full characterization of optimal rates. We study uniform additivity of formulas, which is easily evaluated and captures all known additive quantum formulas. Our complete characterization of uniform additivity exposes an intriguing new additive quantity and identifies a remarkable coincidence-the classical and quantum uniformly additive functions with one auxiliary variable are identical.