S. Engo - Profile on Academia.edu (original) (raw)
Papers by S. Engo
The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou ... more The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou et al. [1] based on quantum matrix with binary decision diagram and nonlinear search algorithm put forth by David Rosenbaum [2] and Abrams and Llyod [3] respectively. Our model that simplifies and generalizes that of Ref. [1], gives the possibility to retrieve one of the sought states in multi-values retrieving, when a measure on the first register is done. If n is the number of qubit of the first register, p ≤ 2n the number of stored patterns, q ≤ p the number of stored patterns if the values of t are known (i.e., t qubits have been measured or are already be disentangled to others, or the oracle acts on a subspace of (n − t) qubits), m ≤ q the number of values x for which f(x) = 1, c = ceil(log2 q) the least integer greater or equal to log2 q, and r = int(log2m) the integer part of log2m, then the time complexity of our algorithm is O(c − r). It is better than Grover’s algorithm and it...
In this paper we present a model of Quantum Associative Memory which can be a helpful tool for ph... more In this paper we present a model of Quantum Associative Memory which can be a helpful tool for physicians without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have similar symptoms. The memory can distinguish single infection from multi-infection. The algorithm used for Quantum Associative Memory is an improve model of original algorithm made by Ventura for Quantum Associative Memory. From the simulation results given, it appears that the efficiency of recognition is good when a particular symptom of a disease with a similar symptoms are inserted.
arXiv: Medical Physics, 2013
In this paper we present a model of Quantum Associative Memory which can be a helpful tool for ph... more In this paper we present a model of Quantum Associative Memory which can be a helpful tool for physicians without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have similar symptoms. The memory can distinguish single infection from multi-infection. The algorithm used for Quantum Associative Memory is an improve model of original algorithm made by Ventura for Quantum Associative Memory. From the simulation results given, it appears that the efficiency of recognition is good when a particular symptom of a disease with a similar symptoms are inserted.
Multistability, staircases, and optical high-order sideband combs in optomechanics
Journal of the Optical Society of America B
Optomechanical systems are known to exhibit self-sustained limit cycles once driven above the par... more Optomechanical systems are known to exhibit self-sustained limit cycles once driven above the parametric instability point. This breaks down the linearized approximation and induces novel nonlinear effects such as dynamical multistability, staircase behavior, and the generation of optical high-order sideband combs (HOSCs). Here, we study the classical nonlinear dynamics of optomechanical systems. We combine numerical simulations and analytical investigation to predict dynamical multistability in the resolved sideband regime. A way to predict the onset of the period doubling process and to control the multistability is analytically provided by tuning the optical linewidth. Indeed, the multistability behavior first changes to a staircase shape and gradually disappears as the system approaches the unresolved sideband limit. We exploit the multistable attractors to generate optical HOSCs by acting solely on the initial values instead of increasing the driving strength. This is the figure of merit of our proposal to relate multistability to the HOSC. As a result, the properties (bandwidth, intensity) of the combs are improved as the mechanical resonator moves towards upper attractors. This work opens a way for low-power HOSC generation in optomechanics and the related technological applications.
Physical Review A
We study steady-state continuous variable entanglement in a three-mode optomechanical system cons... more We study steady-state continuous variable entanglement in a three-mode optomechanical system consisting of an active optical cavity (gain) coupled to a passive optical cavity (loss) supporting a mechanical mode. For a driving laser which is blue-detuned, we show that coupling between optical and mechanical modes is enhanced in the unbroken-PT -symmetry regime. We analyze the stability and this shows that steady-state solutions are more stable in the gain and loss systems. We use these stable solutions to generate distant entanglement between the mechanical mode and the optical field inside the gain cavity. It results in a giant enhancement of entanglement compared to what is generated in the single lossy cavity. This work offers the prospect of exploring quantum state engineering and quantum information in such systems. Furthermore, such entanglement opens up an interesting possibility to study spatially separated quantum objects.
Optomechanical systems close to the conservative limit
Physical Review A, 2017
In dissipative optomechanical systems, the total damping hits negative values at the parametric i... more In dissipative optomechanical systems, the total damping hits negative values at the parametric instability point. This also corresponds to the phonon lasing threshold, where the mechanical resonator enters in the self-induced oscillations regime. This paper shows that the two mentioned phenomena are delayed from each other when the optomechanical systems operate close to their conservative limit, where the mechanical damping is very small. In fact, the total damping can be negative and very small for a while before the phonon lasing happens. As a result, the linearized theory is extended over the negative damping region where the mechanical displacements remain very small. It follows that nonlinear behavior as the dynamical multistability is retarded in such systems.
1/N expansions for central potentials revisted in the light of hypervirial and Hellmann-Feynman theorems and the principle of minimal sensitivity
The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou ... more The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou \etal \cite{zhou2012} based on quantum matrix with binary decision diagram and nonlinear search algorithm put forth by David Rosenbaum \cite{Rosenbaum2010} and Abrams and Llyod \cite{Abrams1998} respectively. Our model that simplifies and generalizes that of Ref. \cite{zhou2012}, gives the possibility to retrieve one of the desired states in multi-values retrieving, when a measure on the first register is done. If nnn is the number of qubit of the first register, pleq2np\leq2^npleq2n the number of stored patterns, qleqpq\leq pqleqp the number of stored patterns if the values of ttt are known (i.e., ttt qubits have been measured or are already be disentangled to others, or the oracle acts on a subspace of (n−t)(n-t)(n−t) qubits), mleqqm\leq qmleqq the number of values xxx for which f(x)=1f(x)=1f(x)=1, c=mathttceil(log_2q)c=\mathtt{ceil}(\log_2 {q})c=mathttceil(log2q) the least integer greater or equal to log2q\log_2{q}log2q, and r=mathttint(log2m)r=\mathtt{int} (\log_2 {m})r=mathttint(log_2m) the integer part ...
arXiv:1405.4483v1 [quant-ph] 18 May 2014, May 18, 2014
We propose a scheme to generate robust stationary continuous-variable entanglement in optomechani... more We propose a scheme to generate robust stationary continuous-variable entanglement in optomechanical systems, based on geometrical nonlinearity that occurs for large mechanical displacements. Such nonlinearity is often used to correct the dynamics of the systems in the strong coupling regime. It appears that geometrical nonlinearity enhances the entanglement and shifts its maximum towards high detuning values. Using the experimental parameters, we find that such a scheme generates a very robust entanglement against thermal decoherence even at room temperature. Our results show that geometrical nonlinearity affects entanglement as the optomechanical quantum interface.
International Journal of Theoretical Physics, 2013
Your article is protected by copyright and all rights are held exclusively by Springer Science+Bu... more Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.
Article
In recent experiments, the re-thermalization time of the mechanical resonator is stated as the li... more In recent experiments, the re-thermalization time of the mechanical resonator is stated as the limiting factor for quantum applications of optomechanical systems. To explain the origin of this limitation, an analytical nonlinear investigation supported by the recent successful experimental laser cooling parameters is carried out in this work. To this end, the effects of geometrical and the optical nonlinearities on the squeezing are studied and are in a good agreement with the experimental results. It appears that highly squeezed state are generated where these nonlinearities are minimized and that high nonlinearities are limiting factors to reach the quantum ground state.
Nonlinearity-induced limitations on cooling in optomechanical systems
Phys. Rev. A 86, 043816 (2012) [6 pages]
"Abstract: In this paper, we apply the technique of back-action cooling to investigate the e... more "Abstract: In this paper, we apply the technique of back-action cooling to investigate the effects of both optical and mechanical nonlinearities in optomechanical cooling systems. It is shown that cooling of the nanomechanical oscillator to its ground state is limited by the effects of these nonlinearities. The qualitative result is justified quantitatively by comparing, for the same parameters, our analytical minimum phonon number with the experimental one."
![Research paper thumbnail of Publisher's Note: Optomechanical systems close to the conservative limit [Phys. Rev. A 95 , 013831 (2017)]](https://attachments.academia-assets.com/83232941/thumbnails/1.jpg)
](Physical Review A
" On page 4, the last sentence of the Acknowledgments should read as "P.D. is grateful to the Aca... more " On page 4, the last sentence of the Acknowledgments should read as "P.D. is grateful to the Academy of Finland for the financial support." The paper has been corrected as of 1 May 2017. The affiliation, footnote and Acknowledgments are incorrect in the printed version of the journal.
Neural Networks
This paper presents the QAMDiagnos, a model of Quantum Associative Memory (QAM) that can be a hel... more This paper presents the QAMDiagnos, a model of Quantum Associative Memory (QAM) that can be a helpful tool for medical staff without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have several similar signs and symptoms. The memory can distinguish a single infection from a polyinfection. Our model is a combination of the improved versions of the original linear quantum retrieving algorithm proposed by Ventura and the non-linear quantum search algorithm of Abrams and Lloyd. From the given simulation results, it appears that the efficiency of recognition is good when particular signs and symptoms of a disease are inserted given that the linear algorithm is the main algorithm. The non-linear algorithm helps confirm or correct the diagnosis or give some advice to the medical staff for the treatment. So, our QAMDiagnos that has a friendly graphical user interface for desktop and smart-phone is a sensitive and a low-cost diagnostic tool that enables rapid and accurate diagnosis of four tropical diseases.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1995
A new analytical expression of a previously proposed Klein-Gordon dipole matrix elements in the q... more A new analytical expression of a previously proposed Klein-Gordon dipole matrix elements in the quasiclassical approach (including quantum defects) is presented.
Relativistic electric dipole radial integrals between Rydberg atomic states in the semiclassical Coulomb approximation
Physics Letters A, 1993
Analytic wave functions are obtained from the Klein-Gordon equation by using the quantum-defect t... more Analytic wave functions are obtained from the Klein-Gordon equation by using the quantum-defect theory in the semiclassical Coulomb approximation. From this result, analytic expressions for relativistic electric dipole radial integrals between highly excited Rydberg states chilambda, chi'lambda' are calculated in terms of Anger functions plus an algebraic part combination of sin (pis) and cos(pis), where s=(sigma'/gs)(chi'-chi)-(sigma/sigma')(chi-chi), chi=sqrt(chichi') . chi, lambda,
Physics Letters A, 1994
A search is conducted for the supersymmetric structure of the Dirac radial equation with the Coul... more A search is conducted for the supersymmetric structure of the Dirac radial equation with the Coulomb potential for spherical coordinates. We derive the relevant features of supersymmetric quantum mechanics for the associated eigenvalues and eigenfunctions. Incorporating symmetry-breaking effects by nonhydrogenic contributions to the effective potential, by using notions of exact quantum-defect theory, we deduce supersymmetry-based quantum-defect eigenvalues and eigenfunctions.
Physics Letters A, 1998
By using the nonrelativistic Good's semiclassical wave functions, in which the quantum defect is ... more By using the nonrelativistic Good's semiclassical wave functions, in which the quantum defect is included, GWKB dipole radial matrix elements are derived in the length and veiocity gauges. The new an~yti~l formulae obtained are expressed in terms of Anger's functions and two new special functions called by us Good's functions. We propose rapidly converging expansion series of these functions, from which numerical values of GWKB radial matrix elements have been computed for comparison with other theoretical results.
Physical Review A, 1997
The comparison of phenomenological and supersymmetry-inspired quantum-defect methods in their rel... more The comparison of phenomenological and supersymmetry-inspired quantum-defect methods in their relativistic and quasirelativistic formulations is undertaken through a systematic study. Derivations, which emphasize similarities and differences between these approaches, are applied to evaluate oscillator strengths for low-lying and Rydberg states for lithiumlike, sodiumlike, and copperlike ions. Relativistic transition matrix elements calculated with the Dirac wave functions are gauge invariant. For the quasirelativistic formalism, an effective transition operator in the velocity gauge is proposed for the calculations. The detailed numerical results we obtained enable us to draw inferences as to the improvements and also to the limitations of these models.
The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou ... more The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou et al. [1] based on quantum matrix with binary decision diagram and nonlinear search algorithm put forth by David Rosenbaum [2] and Abrams and Llyod [3] respectively. Our model that simplifies and generalizes that of Ref. [1], gives the possibility to retrieve one of the sought states in multi-values retrieving, when a measure on the first register is done. If n is the number of qubit of the first register, p ≤ 2n the number of stored patterns, q ≤ p the number of stored patterns if the values of t are known (i.e., t qubits have been measured or are already be disentangled to others, or the oracle acts on a subspace of (n − t) qubits), m ≤ q the number of values x for which f(x) = 1, c = ceil(log2 q) the least integer greater or equal to log2 q, and r = int(log2m) the integer part of log2m, then the time complexity of our algorithm is O(c − r). It is better than Grover’s algorithm and it...
In this paper we present a model of Quantum Associative Memory which can be a helpful tool for ph... more In this paper we present a model of Quantum Associative Memory which can be a helpful tool for physicians without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have similar symptoms. The memory can distinguish single infection from multi-infection. The algorithm used for Quantum Associative Memory is an improve model of original algorithm made by Ventura for Quantum Associative Memory. From the simulation results given, it appears that the efficiency of recognition is good when a particular symptom of a disease with a similar symptoms are inserted.
arXiv: Medical Physics, 2013
In this paper we present a model of Quantum Associative Memory which can be a helpful tool for ph... more In this paper we present a model of Quantum Associative Memory which can be a helpful tool for physicians without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have similar symptoms. The memory can distinguish single infection from multi-infection. The algorithm used for Quantum Associative Memory is an improve model of original algorithm made by Ventura for Quantum Associative Memory. From the simulation results given, it appears that the efficiency of recognition is good when a particular symptom of a disease with a similar symptoms are inserted.
Multistability, staircases, and optical high-order sideband combs in optomechanics
Journal of the Optical Society of America B
Optomechanical systems are known to exhibit self-sustained limit cycles once driven above the par... more Optomechanical systems are known to exhibit self-sustained limit cycles once driven above the parametric instability point. This breaks down the linearized approximation and induces novel nonlinear effects such as dynamical multistability, staircase behavior, and the generation of optical high-order sideband combs (HOSCs). Here, we study the classical nonlinear dynamics of optomechanical systems. We combine numerical simulations and analytical investigation to predict dynamical multistability in the resolved sideband regime. A way to predict the onset of the period doubling process and to control the multistability is analytically provided by tuning the optical linewidth. Indeed, the multistability behavior first changes to a staircase shape and gradually disappears as the system approaches the unresolved sideband limit. We exploit the multistable attractors to generate optical HOSCs by acting solely on the initial values instead of increasing the driving strength. This is the figure of merit of our proposal to relate multistability to the HOSC. As a result, the properties (bandwidth, intensity) of the combs are improved as the mechanical resonator moves towards upper attractors. This work opens a way for low-power HOSC generation in optomechanics and the related technological applications.
Physical Review A
We study steady-state continuous variable entanglement in a three-mode optomechanical system cons... more We study steady-state continuous variable entanglement in a three-mode optomechanical system consisting of an active optical cavity (gain) coupled to a passive optical cavity (loss) supporting a mechanical mode. For a driving laser which is blue-detuned, we show that coupling between optical and mechanical modes is enhanced in the unbroken-PT -symmetry regime. We analyze the stability and this shows that steady-state solutions are more stable in the gain and loss systems. We use these stable solutions to generate distant entanglement between the mechanical mode and the optical field inside the gain cavity. It results in a giant enhancement of entanglement compared to what is generated in the single lossy cavity. This work offers the prospect of exploring quantum state engineering and quantum information in such systems. Furthermore, such entanglement opens up an interesting possibility to study spatially separated quantum objects.
Optomechanical systems close to the conservative limit
Physical Review A, 2017
In dissipative optomechanical systems, the total damping hits negative values at the parametric i... more In dissipative optomechanical systems, the total damping hits negative values at the parametric instability point. This also corresponds to the phonon lasing threshold, where the mechanical resonator enters in the self-induced oscillations regime. This paper shows that the two mentioned phenomena are delayed from each other when the optomechanical systems operate close to their conservative limit, where the mechanical damping is very small. In fact, the total damping can be negative and very small for a while before the phonon lasing happens. As a result, the linearized theory is extended over the negative damping region where the mechanical displacements remain very small. It follows that nonlinear behavior as the dynamical multistability is retarded in such systems.
1/N expansions for central potentials revisted in the light of hypervirial and Hellmann-Feynman theorems and the principle of minimal sensitivity
The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou ... more The model of quantum associative memories proposed here is quietly similar to that of Rigui Zhou \etal \cite{zhou2012} based on quantum matrix with binary decision diagram and nonlinear search algorithm put forth by David Rosenbaum \cite{Rosenbaum2010} and Abrams and Llyod \cite{Abrams1998} respectively. Our model that simplifies and generalizes that of Ref. \cite{zhou2012}, gives the possibility to retrieve one of the desired states in multi-values retrieving, when a measure on the first register is done. If nnn is the number of qubit of the first register, pleq2np\leq2^npleq2n the number of stored patterns, qleqpq\leq pqleqp the number of stored patterns if the values of ttt are known (i.e., ttt qubits have been measured or are already be disentangled to others, or the oracle acts on a subspace of (n−t)(n-t)(n−t) qubits), mleqqm\leq qmleqq the number of values xxx for which f(x)=1f(x)=1f(x)=1, c=mathttceil(log_2q)c=\mathtt{ceil}(\log_2 {q})c=mathttceil(log2q) the least integer greater or equal to log2q\log_2{q}log2q, and r=mathttint(log2m)r=\mathtt{int} (\log_2 {m})r=mathttint(log_2m) the integer part ...
arXiv:1405.4483v1 [quant-ph] 18 May 2014, May 18, 2014
We propose a scheme to generate robust stationary continuous-variable entanglement in optomechani... more We propose a scheme to generate robust stationary continuous-variable entanglement in optomechanical systems, based on geometrical nonlinearity that occurs for large mechanical displacements. Such nonlinearity is often used to correct the dynamics of the systems in the strong coupling regime. It appears that geometrical nonlinearity enhances the entanglement and shifts its maximum towards high detuning values. Using the experimental parameters, we find that such a scheme generates a very robust entanglement against thermal decoherence even at room temperature. Our results show that geometrical nonlinearity affects entanglement as the optomechanical quantum interface.
International Journal of Theoretical Physics, 2013
Your article is protected by copyright and all rights are held exclusively by Springer Science+Bu... more Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.
Article
In recent experiments, the re-thermalization time of the mechanical resonator is stated as the li... more In recent experiments, the re-thermalization time of the mechanical resonator is stated as the limiting factor for quantum applications of optomechanical systems. To explain the origin of this limitation, an analytical nonlinear investigation supported by the recent successful experimental laser cooling parameters is carried out in this work. To this end, the effects of geometrical and the optical nonlinearities on the squeezing are studied and are in a good agreement with the experimental results. It appears that highly squeezed state are generated where these nonlinearities are minimized and that high nonlinearities are limiting factors to reach the quantum ground state.
Nonlinearity-induced limitations on cooling in optomechanical systems
Phys. Rev. A 86, 043816 (2012) [6 pages]
"Abstract: In this paper, we apply the technique of back-action cooling to investigate the e... more "Abstract: In this paper, we apply the technique of back-action cooling to investigate the effects of both optical and mechanical nonlinearities in optomechanical cooling systems. It is shown that cooling of the nanomechanical oscillator to its ground state is limited by the effects of these nonlinearities. The qualitative result is justified quantitatively by comparing, for the same parameters, our analytical minimum phonon number with the experimental one."
Physical Review A
" On page 4, the last sentence of the Acknowledgments should read as "P.D. is grateful to the Aca... more " On page 4, the last sentence of the Acknowledgments should read as "P.D. is grateful to the Academy of Finland for the financial support." The paper has been corrected as of 1 May 2017. The affiliation, footnote and Acknowledgments are incorrect in the printed version of the journal.
Neural Networks
This paper presents the QAMDiagnos, a model of Quantum Associative Memory (QAM) that can be a hel... more This paper presents the QAMDiagnos, a model of Quantum Associative Memory (QAM) that can be a helpful tool for medical staff without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have several similar signs and symptoms. The memory can distinguish a single infection from a polyinfection. Our model is a combination of the improved versions of the original linear quantum retrieving algorithm proposed by Ventura and the non-linear quantum search algorithm of Abrams and Lloyd. From the given simulation results, it appears that the efficiency of recognition is good when particular signs and symptoms of a disease are inserted given that the linear algorithm is the main algorithm. The non-linear algorithm helps confirm or correct the diagnosis or give some advice to the medical staff for the treatment. So, our QAMDiagnos that has a friendly graphical user interface for desktop and smart-phone is a sensitive and a low-cost diagnostic tool that enables rapid and accurate diagnosis of four tropical diseases.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1995
A new analytical expression of a previously proposed Klein-Gordon dipole matrix elements in the q... more A new analytical expression of a previously proposed Klein-Gordon dipole matrix elements in the quasiclassical approach (including quantum defects) is presented.
Relativistic electric dipole radial integrals between Rydberg atomic states in the semiclassical Coulomb approximation
Physics Letters A, 1993
Analytic wave functions are obtained from the Klein-Gordon equation by using the quantum-defect t... more Analytic wave functions are obtained from the Klein-Gordon equation by using the quantum-defect theory in the semiclassical Coulomb approximation. From this result, analytic expressions for relativistic electric dipole radial integrals between highly excited Rydberg states chilambda, chi'lambda' are calculated in terms of Anger functions plus an algebraic part combination of sin (pis) and cos(pis), where s=(sigma'/gs)(chi'-chi)-(sigma/sigma')(chi-chi), chi=sqrt(chichi') . chi, lambda,
Physics Letters A, 1994
A search is conducted for the supersymmetric structure of the Dirac radial equation with the Coul... more A search is conducted for the supersymmetric structure of the Dirac radial equation with the Coulomb potential for spherical coordinates. We derive the relevant features of supersymmetric quantum mechanics for the associated eigenvalues and eigenfunctions. Incorporating symmetry-breaking effects by nonhydrogenic contributions to the effective potential, by using notions of exact quantum-defect theory, we deduce supersymmetry-based quantum-defect eigenvalues and eigenfunctions.
Physics Letters A, 1998
By using the nonrelativistic Good's semiclassical wave functions, in which the quantum defect is ... more By using the nonrelativistic Good's semiclassical wave functions, in which the quantum defect is included, GWKB dipole radial matrix elements are derived in the length and veiocity gauges. The new an~yti~l formulae obtained are expressed in terms of Anger's functions and two new special functions called by us Good's functions. We propose rapidly converging expansion series of these functions, from which numerical values of GWKB radial matrix elements have been computed for comparison with other theoretical results.
Physical Review A, 1997
The comparison of phenomenological and supersymmetry-inspired quantum-defect methods in their rel... more The comparison of phenomenological and supersymmetry-inspired quantum-defect methods in their relativistic and quasirelativistic formulations is undertaken through a systematic study. Derivations, which emphasize similarities and differences between these approaches, are applied to evaluate oscillator strengths for low-lying and Rydberg states for lithiumlike, sodiumlike, and copperlike ions. Relativistic transition matrix elements calculated with the Dirac wave functions are gauge invariant. For the quasirelativistic formalism, an effective transition operator in the velocity gauge is proposed for the calculations. The detailed numerical results we obtained enable us to draw inferences as to the improvements and also to the limitations of these models.