Wave packet construction in three-dimensional quantum billiards: Visualizing the closed orbit, collapse and revival of wave packets in the cubical billiard

We examine the dynamical evolution of wave packets in a cubical billiard where three quantum numbers ($n_x,n_y,n_z$) determine its energy spectrum and consequently its dynamical behavior. We have constructed the wave packet in the cubical billiard and have observed its time evolution for various closed orbits. The closed orbits are possible for certain specific values of quantum numbers ($n_x,n_y,n_z$) and initial momenta ($k_x,k_y,k_z$). We observe that a cubical billiard exhibits degenerate energy levels and the path lengths of the closed orbits for these degenerate energy levels are identical. In spite of the identical path lengths, the shapes of the closed orbits for degenerate levels are different and depend upon angles $\theta$ and $\phi$ which we term as the sweep and the elevation angle respectively. These degenerate levels owe their origin to the symmetries prevailing in the cubical billiard and degenerate levels disappear completely or partially for a parallelepiped billiard as the symmetry breaks due to commensurate or incommensurate ratio of sides.Comment: Kindly contact authors ([email protected]) for graphs and videos of the result

Magic wavelengths for trapping the alkali-metal atoms with circularly polarized light

Extending the search for the magic wavelengths using the circularly polarized light in rubidium [Phys. Rev. A 86, 033416 (2012)], we pursue here to look for the magic wavelengths in the $ns-np_{1/2,3/2}$ transitions of Li, Na and K alkali atoms. These wavelengths for all possible sub-levels are given and compared with the corresponding wavelengths due to the linearly polarized light. We have also mentioned explicitly the dynamic polarizabilities at few important wave lengths. The present study suggests that it is possible to carry out state insensitive trapping of different alkali atoms using the circularly polarized light.Comment: 8 pages, 6 figure

Magnetic sublevel independent magic wavelengths: Application in the Rb and Cs atoms

A generic scheme to trap atoms at the magic wavelengths ($\lambda_{\rm{magic}}$s) that are independent of vector and tensor components of the interactions of the atoms with the external electric field is presented. The $\lambda_{\rm{magic}}$s for the laser cooling D2 lines in the Rb and Cs atoms are demonstrated and their corresponding polarizability values without vector and tensor contributions are given. Consequently, these $\lambda_{\rm{magic}}$s are independent of magnetic sublevels and hyperfine levels of the atomic states involved in the transition, thus, can offer unique approaches to carry out many high precision measurements with minimal systematics. Inevitably, the proposed technique can also be used for electronic or hyperfine transitions in other atomic systems.Comment: Accepted for publication in Phys. Rev.

Dispersion coefficients for the interactions of the alkali and alkaline-earth ions and inert gas atoms with a graphene layer

Largely motivated by a number of applications, the van der Waals dispersion coefficients ($C_3$s) of the alkali ions (Li$^+$, Na$^+$, K$^+$ and Rb$^+$), the alkaline-earth ions (Ca$^+$, Sr$^+$, Ba$^+$ and Ra$^+$) and the inert gas atoms (He, Ne, Ar and Kr) with a graphene layer are determined precisely within the framework of Dirac model. For these calculations, we have evaluated the dynamic polarizabilities of the above atomic systems very accurately by evaluating the transition matrix elements employing relativistic many-body methods and using the experimental values of the excitation energies. The dispersion coefficients are, finally, given as functions of the separation distance of an atomic system from the graphene layer and the ambiance temperature during the interactions. For easy extraction of these coefficients, we give a logistic fit to the functional forms of the dispersion coefficients in terms of the separation distances at the room temperature.Comment: 5 figures, 2 table

Dispersion C3 coefficients for the alkali-metal atoms interacting with a graphene layer and with a carbon nanotube

We evaluate separation dependent van der Waal dispersion ($C_3$) coefficients for the interactions of the Li, Na, K and Rb alkali atoms with a graphene layer and with a single walled carbon nanotube (CNT) using the hydrodynamic and Dirac models. The results from both the models are evaluated using accurate values of the dynamic polarizabilities of the above atoms. Accountability of these accurate values of dynamical polarizabilities of the alkali atoms in determination of the above $C_3$ coefficients are accentuated by comparing them with the coefficients evaluated using the dynamic dipole polarizabilities estimated from the single oscillator approximation which are typically employed in the earlier calculations. For practical description of the atom-surface interaction potentials the radial dependent $C_3$ coefficients are given for a wide range of separation distances between the ground states of the considered atoms and the wall surfaces and also for different values of nanotube radii. The coefficients for the graphene layer are fit to a logistic function dependent on the separation distance. For CNT, we have carried out a paraboloid kind of fit dependent on both the separation distances and radii of the CNT. These fitted functions,with the list of fitting parameters, can be used to extrapolate the interaction potentials between the considered alkali atoms and the graphene layer or CNT surface conveniently at the given level of accuracy.Comment: 9 pages, 3 figure

Dispersion coefficients for the interaction of Cs atom with different material media

Largely motivated by a number of applications, the dispersion ($C_3$) coefficients for the interaction of a Cs atom with different material media such as Au (metal), Si (semiconductor) and various dielectric surfaces like SiO$_2$, SiN$_{\rm{x}}$, sapphire and YAG are determined using accurate values of the dynamic polarizabilities of the Cs atom obtained employing the relativistic coupled-cluster approach and the dynamic dielectric constants of the walls. Moreover, we also give the retardation coefficients in the graphical representation as functions of separation distances to describe the interaction potentials between the Cs atom with the above considered material media. For the easy access to the interaction potentials at a given distance of separation, we devise a simple working functional fitting form for the retarded coefficients in terms of two parameters that are quoted for each medium.Comment: 3 figures, 5 table

Multipolar Black Body Radiation Shifts for the Single Ion Clocks

Appraising the projected $10^{-18}$ fractional uncertainty in the optical frequency standards using singly ionized ions, we estimate the black-body radiation (BBR) shifts due to the magnetic dipole (M1) and electric quadrupole (E2) multipoles of the magnetic and electric fields, respectively. Multipolar scalar polarizabilities are determined for the singly ionized calcium (Ca$^+$) and strontium (Sr$^+$) ions using the relativistic coupled-cluster method; though the theory can be exercised for any single ion clock proposal. The expected energy shifts for the respective clock transitions are estimated to be $4.38(3) \times 10^{-4}$ Hz for Ca$^+$ and $9.50(7) \times 10^{-5}$ Hz for Sr$^+$. These shifts are large enough and may be prerequisite for the frequency standards to achieve the foreseen $10^{-18}$ precision goal.Comment: 1 figure, 4 table

Comparing magic wavelengths for the 6s 2S1/2−6p 2P1/2,3/26s ~ {^2}S_{1/2}-6p ~ {^2}P_{1/2,3/2} transitions of Cs using circularly and linearly polarized light

We demonstrate magic wavelengths, at which external electric field produces null differential Stark shifts, for the $6s ~ {^2}S_{1/2}-6p ~ {^2}P_{1/2,3/2}$ transitions in the Cs atom due to circularly polarized light. In addition, we also obtain magic wavelengths using linearly polarized light, in order to verify the previously reported values, and make a comparative study with the values obtained for circularly polarized light. A number of these wavelengths are found to be in the optical region and could be of immense interest to experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, $6p ~{^2}P_{1/2}$ and $6p ~{^2}P_{3/2}$ states of Cs. We use the available precise values of the electric dipole (E1) matrix elements of the transitions that give the dominant contributions from the lifetime measurements of the excited states. Other significantly contributing E1 matrix elements are obtained by employing a relativistic coupled-cluster singles and doubles method. The accuracies of the dynamic polarizabilities are substantiated by comparing the static polarizability values with the corresponding experimental results.Comment: 9 pages, 4 figure

Static and dynamic polarizabilities of Yb-ion]{Accurate determination of black-body radiation shift, magic and tune-out wavelengths for the 6S1/2→5D3/2\rm 6S_{1/2} \rightarrow 5D_{3/2} clock transition in Yb+^+

We present precise values of the dipole polarizabilities ($\alpha$) of the ground $\rm [4f^{14}6s] ~ ^2S_{1/2}$ and metastable $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ states of Yb$^+$, that are %vital {\bf important} in reducing systematics in the clock frequency of the $\rm[4f^{14}6s] ~ ^2S_{1/2} \rightarrow [4f^{14}5d] ~ ^2D_{3/2}$ transition. The static values of $\alpha$ for the ground and $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ states are estimated to be $9.8(1) \times 10^{-40} \,\,\rm Jm^2V^{-2}$ and $17.6(5) \times 10^{-40}\,\, \rm Jm^2V^{-2}$, respectively, while the tensor contribution to the $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ state as $- 12.3(3) \times 10^{-40}\,\, \rm Jm^2V^{-2}$ compared to the experimental value $-13.6(2.2) \times 10^{-40}\,\,\rm Jm^2V^{-2}$. This corresponds to the differential scalar polarizability value of the above transition as $-7.8$(5)$\,\times\, 10^{-40}\,\rm Jm^2 V^{-2}$ in contrast to the available experimental value $-6.9$(1.4)$\,\times\, 10^{-40}$\,\, $\rm Jm^2V^{-2}$. This results in the black-body radiation (BBR) shift of the clock transition as $-0.44(3)$ Hz at the room temperature, which is large as compared to the previously estimated values. Using the dynamic $\alpha$ values, we report the tune-out and magic wavelengths that could be of interest to subdue %major systematics due to the Stark shifts and for constructing lattice optical clock using Yb$^+$.Comment: 18 pages, 3 figure

Emending thermal dispersion interactions of Li, Na, K and Rb alkali metal-atoms with graphene in the Dirac model

Using accurate dynamic polarizabilities of Li, Na, K and, Rb atoms, we scrutinize the thermal Casimir-Polder interactions of these atoms with a single layered graphene. Considering the modified Lifshitz theory for material interactions, we reanalyze the dispersion coefficients ($C_3$s) of the above atoms with graphene as functions of separation distance, gap parameter and temperature among which some of them were earlier studied by estimating dynamic polarizabilities of the above atoms from the single oscillator model approximation. All these $C_3$ coefficients have been evaluated in the framework of the Dirac model. The interactions are described for a wide range of distances and temperatures to demonstrate the changes in behavior with the varying conditions of the system and also sensitivities in the interactions are analyzed by calculating them for different values of the gap parameter. From these analyses, we find a suitable value of the gap parameter for which the true nature of the interactions in graphene can be surmised more accurately.Comment: 8 pages, 7 figures, 1 tabl