Large Quantum Numbers Research Articles

Electric dipole polarizabilities of Rydberg states of alkali-metal atoms

Calculations of the static electric-dipole scalar and tensor polarizabilities are presented for two alkali atoms, Rb and Cs, for the $nS$, $nP_{1/2, 3/2}$, and $nD_{3/2, 5/2}$ states with large principal quantum numbers up to $n = 50$. The calculations are performed within an effective one-electron approximation, based on the Dirac-Fock Hamiltonian with a semi-empirical core-polarization potential. The obtained results are compared with those from a simpler semi-empirical approach and with available experimental data.

Revealing a quantum feature of dimensionless uncertainty in linear and quadratic potentials by changing potential intervals

As an undergraduate exercise, in an article (2012 Am. J. Phys. 80 780–14), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on and the number of states (n), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding . But the question is whether the uncertainty still remains dependent on quantum number n. In the above-mentioned article, there lies this contrast; on the one hand, the dimensionless quantum uncertainty of the potential box approaches classical dispersion only in the limit of large quantum numbers ()—consistent with the correspondence principle. On the other hand, similar evaluations for bouncing ball and harmonic oscillator potentials are equal to their classical counterparts independent of n. This equality may hide the quantum feature of low energy levels. In the current study, we change the potential intervals in order to make them symmetric for the linear potential and non-symmetric for the quadratic potential. As a result, it is shown in this paper that the dimensionless quantum uncertainty of these potentials in the new potential intervals is expressed in terms of quantum number n. In other words, the uncertainty requires the correspondence principle in order to approach the classical limit. Therefore, it can be concluded that the dimensionless analysis, as a useful pedagogical method, does not take away the quantum feature of the n-dependence of quantum uncertainty in general. Moreover, our numerical calculations include the higher powers of the position for the potentials.

Spectrum, radial wave functions, and hyperfine splittings of the Rydberg states in heavy alkali-metal atoms

We calculate the bound state spectrum of the highly excited valence electron in the heavy alkali atoms solving the radial Schr\"odinger eigenvalue problem numerically with an accurate spectral collocation algorithm that applies also for a large principal quantum number $n\gg1$. As an effective single-particle potential we favor the reputable potential of Marinescu \emph{et al}., {[}Phys. Rev.A \textbf{49}, 982(1994){]}. Recent quasiclassical calculations of the quantum defect of the valence electron agree for orbital angular momentum $l=0,1,2,...$ overall remarkably well with the results of the numerical calculations, but for the Rydberg states of rubidium and also cesium with $l=3$ this agreement is less fair. The reason for this anomaly is that the potential acquires for $l=3$ deep inside the ionic core a tiny second classical region, thus invalidating a standard WKB calculation with two widely spaced turning points. Comparing then our numerical solutions of the radial Schr\"odinger eigenvalue problem with the uniform analytic WKB approximation of Langer we observe everywhere a remarkable agreement, apart from a tiny region around the inner turning point. With help of an ansatz proposed by Fock we obtain for the \emph{s}-states a second uniform analytic approximation to the radial wave function complementary to the WKB approximation of Langer, which is exact for $r\to0^{+}$. The value of the radial \emph{s}-wave function at $r=0$ is analytically found, thus validating the Fermi-Segr\`e formula for the magnetic dipole interaction constant $A_{n,j,0}^{\left(\mathrm{HFS}\right)}$.

On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity

Building on a recent proposal for a quantum reduction to spherical symmetry from full loop quantum gravity, we investigate the relation between a quantisation of spherically symmetric general relativity and a reduction at the quantum level. To this end, we generalise the previously proposed quantum reduction by dropping the gauge fixing condition on the radial diffeomorphisms, thus allowing us to make direct contact with previous work on reduced quantisation. A dictionary between spherically symmetric variables and observables with respect to the reduction constraints in the full theory is discussed, as well as an embedding of reduced quantum states to a subsector of the quantum symmetry reduced full theory states. On this full theory subsector, the quantum algebra of the mentioned observables is computed and shown to qualitatively reproduce the quantum algebra of the reduced variables in the large quantum number limit for a specific choice of regularisation. Insufficiencies in recovering the reduced algebra quantitatively from the full theory are attributed to the oversimplified full theory quantum states we use.

Thermodynamic properties of frustrated arbitrary spin-S J1–J2 quantum Heisenberg antiferromagnet on the body-centered-cubic lattice in random phase approximation

Abstract The thermodynamic properties of the frustrated arbitrary spin-S J1–J2 Heisenberg antiferromagnet on the body-centered-cubic lattice for Neel phase are systematically calculated by use of the double-time Green׳s function method within the random phase approximation (RPA). The role of spin quantum number and frustration strength on sublattice magnetization, Neel temperature, internal energy, and free energy are carefully analyzed. The curve of zero-temperature sublattice magnetization 〈 S z ( 0 ) 〉 / S versus frustration strength J2/J1 values are almost flat at the larger spin quantum number S=10. With the increase of normalized temperature T/TN, the larger the spin quantum number S, the faster the 〈 S z 〉 / S drops, and the smaller influence of J2/J1 on the 〈 S z 〉 / S versus T/TN curve. Under the RPA approach, the Neel temperature TN /Sp and the internal energy E/Sp at the Neel point are independent of spin quantum number S. The numerical results show that the internal energy E/Sp at the Neel point seems independent of the frustration strength J2/J1. This indicates that thermodynamic quantities have universal characteristics for large spin quantum number.

Spin–charge separation of dark-state polaritons in a Rydberg medium

The propagation of light fields through a quasi one-dimensional cold atomic gas, exciting atomic Rydberg levels of large principal quantum number under conditions of electromagnetically induced transparency, can lead to a stable two-mode Luttinger liquid system. Atomic van der Waals interactions induce a coupling of bosonic field modes that display both photonic and atomic character, the Rydberg dark-state polaritons (RDPs). It is shown that by tunable control of the van der Waals coupling, the RDP may decouple into independent ‘spin’ and ‘charge’ fields which propagate at different speeds, analogous to spin–charge separation of electrons in a one-dimensional metal.

Bekenstein–Hawking entropy by energy quantization from Reissner–Nordström black hole

We consider the motion of a test particle orbiting around Reissner–Nordström (RN) black hole spacetime. The complete set of equations for radial motion and effective potential is derived. We also derive the radius of the different stable circular orbits of this particle corresponding to different label indexes like the Bohr atomic model. We also quantized the energy of this particle from the quantization of angular momentum and calculated the Bekenstein–Hawking entropy of RN black hole. We also investigate the change of entropy between two nearby states approaches to zero for large quantum numbers.

Contraction limits of the proton-neutron symplectic model

The algebraic approach to nuclear structure physics allows a certain microscopic collective motion algebra to be also interpreted on macroscopic level which is achieved in the limit of large representation quantum numbers. Such limits are referred to as macroscopic or hydrodynamic limits and show how a given microscopic discrete system starts to behave like a continuous fluid. In the present paper, two contraction limits of the recently introduced fully microscopic proton-neutron symplectic model (PNSM) with the Sp(12; R) dynamical symmetry algebra are considered. As a result, two simplified macroscopic models of nuclear collective motion are obtained in simple geometrical terms. The first one is the U(6)-phonon model with the semi-direct product structure [HW(21)]U(6), which is shown to be actually an alternative formulation of the original proton-neutron symplectic model in the familiar IBM-terms. The second model which appears in double contraction limit is the two-rotor model with the ROTp(3) ⊗ ROTn(3) ⊃ ROT(3) algebraic structure. The latter, in contrast to the original two-rotor model, is not restricted to the case of two coupled axial rotors. In this way, the second contraction limit of the PNSM, provides the phenomenological two-rotor model with a simple microscopic foundation.

Electromagnetically induced transparency of Rydberg atoms in modulated laser fields

Rydberg atoms, with large principal quantum number n, have been widely investigated in recent years due to their peculiar properties, such as big sizes, long lifetimes and strong interactions with fields and other Rydberg atoms. Rydberg atoms are very sensitive to external fields due to their large polarizabilities scaling as n7.These make Rydberg atoms an ideal candidate for the quantum information, the many-body interaction, etc.In this work, we investigate the Rydberg atoms using electromagneticlly induced transparency (EIT) in a ladder three-level system. The EIT is a quantum interference effect between two excitation path-ways driven by two laser fields. The main idea is performed in a room temperature cesium vapor cell, where the probe laser frequency is modulated. The ground state (6P1/2), excited state (6P3/2), and Rydberg state (nS1/2) constitute a Rydberg three-level system, in which the probe laser is fixed to the 6S1/2 (F = 4)6P3/2 (F = 5) transition by saturated absorption spectrum technique, whereas the coupling laser is scanned across the 6P3/249S1/2 transition. We detect the demodulated EIT signal with the lock-in amplifier (SR830). The modulated EIT signal shows a two-peak structure. The measured spacing between two peaks increases with the frequency detuning, caused by the modulation amplitude, and half the spacing between the peak-to-peak is nearly 1.67 times the modulation amplitude of the probe laser; the measured result shows that the splitting is independent of the modulation frequency. The experimental results are in agreement with the theoretical calculations. The results in our work can be used for real-time monitoring of the laser-line profiles and the fluctuation of laser frequency.

Measurements of the number density of water molecules in plasma by using a combined spectral−probe method

A novel method for measuring the number density of water molecules in low-temperature plasma is developed. The absolute intensities of rotational lines of the (0,0) band of the OH(A2Σ–X2П) transition are used. Lines with sufficiently large rotational quantum numbers referring to the so-called “hot” group of molecules produced by electron-impact dissociative excitation of water molecules are chosen for measurements. The excitation rate of a process with a known cross section is determined by measuring the parameters of plasma electrons by means of the probe method. The measured number densities of molecules are compared with those in the initial plasma-forming mixture. The time evolution of the particle densities in plasma is investigated. The problems of the sensitivity and applicability of the absolute spectral method are considered.

Bekenstein–Hawking Entropy by energy quantization from Schwarzschild-de Sitter black hole

The energy of a test particle orbiting a Schwarzschild-de Sitter (SdS) black hole is quantized with the help of the quantization of angular momentum. We have investigated the change of entropy between two nearby states using energy quantization. We have shown that the change of entropy as well as purely thermal emission rate is dependent of quantum number and approach to zero for large quantum number.

Hydrogen-Like Atom in Spaces of Lower Dimensions

The problem of finding discrete energy values of a particle with negative charge equal in absolute value to the elementary charge in one-dimensional space (e → e 1), located in the field of a nucleus with charge (Ze 1) with potential corresponding to the space of this dimension (N = 1) and different from the potential of the nucleus in three-dimensional space (N = 3) is solved in the quasiclassical approximation. For the one-dimensional case, the corresponding Schrodinger equation is solved and exact energy values are obtained, coincident with the quasiclassical approximation in the limit of large quantum numbers, and the wave function, expressed in terms of the Airy function, is found. In this latter approach, the energy values depend on the zeros of the Airy function. Considerations are discussed, touching on the possibility of solution of the Schrodinger equation for a hydrogen-like atom in two-dimensional space (N = 2).

Dynamically self-regular quantum harmonic black holes

The recently proposed UV self-complete quantum gravity program is a new and very interesting way to envision Planckian/trans-Planckian physics. In this new framework, high energy scattering is dominated by the creation of micro black holes, and it is experimentally impossible to probe distances shorter than the horizon radius. In this letter we present a model which realizes this idea through the creation of self-regular quantum black holes admitting a minimal size extremal configuration. Their radius provides a dynamically generated minimal length acting as a universal short-distance cutoff. We propose a quantization scheme for this new kind of microscopic objects based on a Bohr-like approach, which does not require a detailed knowledge of quantum gravity. The resulting black hole quantum picture resembles the energy spectrum of a quantum harmonic oscillator. The mass of the extremal configuration plays the role of zero-point energy. Large quantum number re-establishes the classical black hole description. Finally, we also formulate a “quantum hoop conjecture” which is satisfied by all the mass eigenstates and sustains the existence of quantum black holes sourced by Gaussian matter distributions.

Magic state distillation and gate compilation in quantum algorithms for quantum chemistry

Quantum algorithms for quantum chemistry map the dynamics of electrons in a molecule to the dynamics of a coupled spin system. To reach chemical accuracy for interesting molecules, a large number of quantum gates must be applied which implies the need for quantum error correction and fault‐tolerant quantum computation. Arbitrary fault‐tolerant operations can be constructed from a small, universal set of fault‐tolerant operations by gate compilation. Quantum chemistry algorithms are compiled by decomposing the dynamics of the coupled spin‐system using a Trotter formula, synthesizing the decomposed dynamics using Clifford operations and single‐qubit rotations, and finally approximating the single‐qubit rotations by a sequence of fault‐tolerant single‐qubit gates. Certain fault‐tolerant gates rely on the preparation of specific single‐qubit states referred to as magic states. As a result, gate compilation and magic state distillation are critical for solving quantum chemistry problems on a quantum computer. We review recent progress that has improved the efficiency of gate compilation and magic state distillation by orders of magnitude. © 2015 Wiley Periodicals, Inc.

Two-photon radio frequency spectroscopy of 49S Rydberg state

Rydberg atoms, with large principal quantum number, exhibit certain properties, such as long lifetimes and strong interactions with fields and other atoms, which have been extensively investigated recently. One of the properties is the electromagnetically induced transparency (EIT) of Rydberg ladder system, which can be used to measure the radio frequency (RF) field with high sensitivity. In this paper, we investigate the quantum coherent effect of cesium Rydberg atom in a three-level ladder system involving the ground state (6S1/2), the excited state (6P3/2) and 49S1/2 Rydberg state in room temperature vapor cell. The probe laser (852 nm) drives the transition of 6S1/2(F=4)→6P3/2(F'=5), while the coupling laser (510 nm) couples the Rydberg transition of 6P3/2 (F'=5)→nS1/2. A typical electromagnetically induced transparency spectrum is obtained when the weak probe laser is scanned through the transition of 6S1/2(F=4)→6P3/2(F'=5) and the coupling laser tuning to Rydberg transition. The two-photon RF spectra are observed when the RF field with a frequency of ～16.9 GHz couples the Rydberg transition of 49S1/2→47D3/2, where the EIT signal is split into two EIT peaks due to the interaction between the RF field and Rydberg atoms. The dependences of EIT splitting on the power of RF field are investigated. The results show that the EIT splitting increases with the power of RF field, which can inversely be used to measure the RF field with a higher spatial resolution in the future.

Potentials of long-range cesium Rydberg molecule

Rydberg atom, with a large principal quantum number n, has big size, long lifetime, strong long-range interactions, and so on. These properties make Rydberg atoms potential candidate of quantum gate and single-photon source. Rydberg electron can interact with nearby ground-state atom, which is polarized by the Rydberg electron and is bound to the orbit of Rydberg electrons forming Rydberg molecule. As the kinetic energy of the Rydberg electron is very low, only the lowest partial waves will contribute to the molecular potential.#br#In this paper, the low electron-atom scattering with the semi-classical approximation is introduced, and the pseudopotential of interaction between Rydberg electron and ground-state atom is used to describe the long-range Rydberg molecular potential. Molecular potential curves for cesium (nS, n=30-60) are plotted according to the results of numerical computation, from which the outermost potential depth De and the equilibrium distance r0 of long-range cesium Rydberg molecule are deduced. Potential curves of cesium Rydberg molecules are consistent with the distribution curves in radial probability densities of cesium Rydberg electrons. Dependences of De and r0 on the principal quantum number n are investigated, this has an important role for the experimental measurements. The size of a Rydberg molecule depends on the equilibrium distance r0 and is proportional to the square of effective principal quantum number (n-δ )2. The calculated outermost potential depth De of Rydberg molecule becomes smaller with the increase of principal quantum number n. Rydberg molecule is very sensitive to the external field and can be used to measure and monitor weak signals.

Controlling the interactions of a few cold Rb Rydberg atoms by radio-frequency-assisted Förster resonances

Long-range interactions between cold Rydberg atoms, which are used in many important applications, can be enhanced using F\"orster resonances between collective many-body states controlled by an external electric field. Here we report on the first experimental observation of highly-resolved radio-frequency-assisted F\"orster resonances in a few cold Rb Rydberg atoms. We also observed radio-frequency-induced F\"orster resonances which cannot be tuned by a dc electric field. They imply an efficient transition from van der Waals to resonant dipole-dipole interaction due to Floquet sidebands of Rydberg levels appearing in the rf-field. This method can be applied to enhance the interactions of almost arbitrary Rydberg atoms with large principal quantum numbers.

Giant Rydberg excitons in the copper oxide Cu2O.

A highly excited atom having an electron that has moved into a level with large principal quantum number is a hydrogen-like object, termed a Rydberg atom. The giant size of Rydberg atoms leads to huge interaction effects. Monitoring these interactions has provided insights into atomic and molecular physics on the single-quantum level. Excitons--the fundamental optical excitations in semiconductors, consisting of an electron and a positively charged hole--are the condensed-matter analogues of hydrogen. Highly excited excitons with extensions similar to those of Rydberg atoms are of interest because they can be placed and moved in a crystal with high precision using microscopic energy potential landscapes. The interaction of such Rydberg excitons may allow the formation of ordered exciton phases or the sensing of elementary excitations in their surroundings on a quantum level. Here we demonstrate the existence of Rydberg excitons in the copper oxide Cu2O, with principal quantum numbers as large as n = 25. These states have giant wavefunction extensions (that is, the average distance between the electron and the hole) of more than two micrometres, compared to about a nanometre for the ground state. The strong dipole-dipole interaction between such excitons is indicated by a blockade effect in which the presence of one exciton prevents the excitation of another in its vicinity.

The Lifshitz-Kosevich-Shoenberg theory of relativistic electronic gas in neutron stars

Similar to the de Haas-van Alphen magnetic oscillatory in some normal metals when the Landau quantization is predominant, the magnetic oscillation can also occur in highly degenerate and relativistic electron gas in neutron stars. At large Landau quantum number (Landau quantum number r≥2), we generalize the Lifshitz-Kosevich-Shoenberg theory in non-relativistic electron gas to relativistic gas. At small Landau quantum number (r<2), we expand the grand potential into Fourier series and get similar harmonic oscillatory formula of magnetization. These results indicate that magnetic phase transition similar as Condon transition observed in metals can appear in neutron stars when the differential susceptibility exceeds 1/4π.

On the calculation of line strengths, oscillator strengths and lifetimes for very large principal quantum numbers in hydrogenic atoms and ions by the McLean–Watson formula

As a sequel to an earlier study (Hey 2009 J. Phys. B: At. Mol. Opt. Phys. 42 125701), we consider further the application of the line strength formula derived by Watson (2006 J. Phys. B: At. Mol. Opt. Phys. 39 L291) to transitions arising from states of very high principal quantum number in hydrogenic atoms and ions (Rydberg–Rydberg transitions, n > 1000). It is shown how apparent difficulties associated with the use of recurrence relations, derived (Hey 2006 J. Phys. B: At. Mol. Opt. Phys. 39 2641) by the ladder operator technique of Infeld and Hull (1951 Rev. Mod. Phys. 23 21), may be eliminated by a very simple numerical device, whereby this method may readily be applied up to n ≈ 10 000. Beyond this range, programming of the method may entail greater care and complexity. The use of the numerically efficient McLean–Watson formula for such cases is again illustrated by the determination of radiative lifetimes and comparison of present results with those from an asymptotic formula. The question of the influence on the results of the omission or inclusion of fine structure is considered by comparison with calculations based on the standard Condon–Shortley line strength formula. Interest in this work on the radial matrix elements for large n and n′ is related to measurements of radio recombination lines from tenuous space plasmas, e.g. Stepkin et al (2007 Mon. Not. R. Astron. Soc. 374 852), Bell et al (2011 Astrophys. Space Sci. 333 377), to the calculation of electron impact broadening parameters for such spectra (Watson 2006 J. Phys. B: At. Mol. Opt. Phys. 39 1889) and comparison with other theoretical methods (Peach 2014 Adv. Space Res. in press), to the modelling of physical processes in H II regions (Roshi et al 2012 Astrophys. J. 749 49), and the evaluation bound–bound transitions from states of high n during primordial cosmological recombination (Grin and Hirata 2010 Phys. Rev. D 81 083005, Ali-Haïmoud and Hirata 2010 Phys. Rev. D 82 063521, Ali-Haïmoud 2013 Phys. Rev. D 87 023526).