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arXiv:1704.08211 · Physical Review A

Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis

Amine Jaouadi, R. Lefèbvre, Osman Atabek

Published Jun 13, 2017• 1 citation
arXivHTMLPDF

Abstract

A semiclassical model supporting the destructive interference interpretation of zero-width resonances (ZWRs) is extended to wavelengths inducing ${c}^{\ensuremath{-}}$-type curve crossing situations in ${\mathrm{Na}}_{2}$ strong-field dissociation. This opens the possibility to get critical couples of wavelengths $\ensuremath{\lambda}$ and field intensities $I$ to reach ZWRs associated with the ground vibrationless level $v=0$, that, contrary to other vibrational states ($v>0$), is not attainable for the commonly referred ${c}^{+}$-type crossings. The morphology of such ZWRs in the laser $(I,\ensuremath{\lambda})$ parameter plane and their usefulness in filtration strategies aiming at molecular cooling down to the ground $v=0$ state are examined within the frame of an adiabatic transport scheme.

DOI
10.1103/physreva.95.063409

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Analytical Treatment of Some Quantum Transitions in Molecular Dynamic Processes

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They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. 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On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. 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This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.

•Sep 29, 2000•0 citations
A semiclassical model supporting the destructive interference interpretation of zero-width resonances (ZWRs) is extended to wavelengths inducing c\ensuremath−{c}^{\ensuremath{-}}c\ensuremath−-type curve crossing situations in Na2{\mathrm{Na}}_{2}Na2​ strong-field dissociation. This opens the possibility to get critical couples of wavelengths \ensuremathλ\ensuremath{\lambda}\ensuremathλ and field intensities III to reach ZWRs associated with the ground vibrationless level v=0v=0v=0, that, contrary to other vibrational states (v>0v>0v>0), is not attainable for the commonly referred c+{c}^{+}c+-type crossings. The morphology of such ZWRs in the laser (I,\ensuremathλ)(I,\ensuremath{\lambda})(I,\ensuremathλ) parameter plane and their usefulness in filtration strategies aiming at molecular cooling down to the ground v=0v=0v=0 state are examined within the frame of an adiabatic transport scheme.
In molecular photodissociation, some specific combinations of laser parameters (wavelength and intensity) lead to unexpected zero-width resonances (ZWRs) with, in principle, infinite lifetimes. Their potential to induce basic quenching mechanisms has recently been devised in the laser control of vibrational cooling through filtration strategies [O. Atabek et al., Phys. Rev. A 87, 031403(R) (2013)]. A full quantum adiabatic control theory based on the adiabatic Floquet Hamiltonian is developed to show how a laser pulse could be envelope-shaped and frequency-chirped so as to protect a given initial vibrational state against dissociation, taking advantage of its continuous transport on the corresponding ZWR all along the pulse duration. As compared with previous control scenarios that actually suffered from nonadiabatic contamination, drastically different and much more efficient filtration goals are achieved. A semiclassical analysis helps us to find and interpret a complete map of ZWRs in the laser parameter plane. In addition, the choice of a given ZWR path, among the complete series identified by the semiclassical approach, turns out to be crucial for the cooling scheme, targeting a single vibrational state population left at the end of the pulse, while all others have almost completely decayed. The illustrative example, which has the potential to be transposed to other diatomics, is Na2{\mathrm{Na}}_{2}Na2​ prepared by photoassociation in vibrationally hot but translationally and rotationally cold states.
Feshbach resonances play a major role in translationally cold-molecule preparation. In this context, their laser control is of crucial importance. This work is devoted to the depiction of some basic mechanisms of such a control using intense, short laser pulses and referring to nonlinear multiphoton processes. Our goal is to adiabatically transport a Feshbach resonance onto a zero-width resonance, the characteristics of which have already been discussed in the literature. Three processes are then addressed: (i) during the rise of the pulse and its plateau, the preparation of a so-called laser bound molecule (LBM) still stable, but structurally different from the standard chemically bound molecule; (ii) during the pulse switching off, an adiabatic transport of this LBM on a very few excited vibrational levels, and (iii) concomitantly, a filtration strategy to photodissociate all these levels except one, giving thus rise to but a single field-free excited vibrational state. With or without an eventual stimulated Raman adiabatic passage technique to bring all the population to the ground rovibrational state, this opens an alternate for a full optical control of ultracold-molecule formation. The illustrative example, offering the potentiality to be transposed to other diatomics, is H2+{\mathrm{H}}_{2}^{+}H2+​.
A zero-width resonance (ZWR) is also called a bound state in a continuum. Such resonances occur when the Floquet method is applied to a diatomic molecule submitted to a continuous wave field. This circumstance requires particular choices of wavelength and electric field amplitude. The corresponding wave function can be factorised with a method devised for bound states. The result is a product of an electronic factor depending parametrically on the nuclear coordinate by a nuclear factor. The factorisation of the wave function is done through the introduction of a fictitious time-independent Hamiltonian yielding the same coupled equations as the Floquet algorithm applied to the time-dependent Hamiltonian. The potential which determines the nuclear factor is calculated and used to recover the real ZWR quasi-energy. Application is made to the molecule Na2 prepared by photoassociation in the metastable 3Σ+ u state. Photodissociation of this state by laser-coupling to the (1)3Π g state requires an intensity low enough to require the consideration of only two channels (one photon interaction) in the application of the Floquet method.
Exceptional points (EPs) corresponding to resonance coalescence (i.e., complex energy degeneracy and identical wave functions) occur in many areas of non-Hermitian physics and, in particular, in laser-induced molecular dynamics for specific choices of two control parameters. We have previously shown [Atabek, et al., Phys. Rev. Lett. 106, 173002 (2011)] how these control parameters, namely, the wavelength and the intensity of the external field, have to be tuned to take advantage of EPs for selective vibrational transfers within a reduced one-dimensional model describing a diatomic molecule with frozen rotation. Moreover, the possibility offered by such transfers to adiabatically transport all the vibrational population to the ground v=0v=0v=0 level has been presented as a realistic vibrational cooling strategy with an entropy flow toward the field-induced dissociative channel. The purpose of the present article is twofold: (i) Extend the model to a full three-dimensional quantum description of the rotating molecule and discuss the existence, determination, and role of EPs involving rovibrational resonances; (ii) examine the possibility for a further challenging step in obtaining ultracold molecules through combined vibrational and rotational laser control dynamics, aiming at total purification for reaching the ground (v=J=0v=J=0v=J=0) rovibrational level. The illustrative example is the Na2{\mathrm{Na}}_{2}Na2​ molecule for which translationally cold species have experimentally been obtained.
Laser control of complete vibrational transfer in Na2_22​ using resonance coalescence
With a specific choice of laser parameters resulting into a so-called exceptional point in the wavelength-intensity plane, it is possible to produce the coalescence of two Floquet resonances describing the photodissociation of the molecule Na2_22​, which is one of the candidates for molecular cooling. Appropriately tuning laser parameters, following a contour around the exceptional point, the resonances exchange their labels. This represents a laser control of the vibrational transfer from one field-free state to another, through an adiabatic transport involving these resonances. The proportion of undissociated molecules at the end of the pulse is checked through Floquet adiabatic theory. A vibrational cooling scenario can be proposed based on a complete vibrational transfer which is predicted, with only 20 percent of molecules undergoing dissociation.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.