Two-photon Transition Probability Research Articles

Two-photon magneto-optical transitions in quantum rings

We study the effect of the magnetic and Aharonov–Bohm (AB) flux fields on the two-photon absorption (2PA) in a quantum ring (QR) system. The expression of 2PA coefficient is expressed through the two-photon transition probability including both intraband and interband transitions. The effect of the polarization orientation of the incident light breaks the symmetry from the contribution between the AB flux field and the azimuthal quantum number. Magnetic and AB flux fields have a significant effect on the electronic properties of the QR. The 2PA spectra for intraband transitions are in the THz range and have a much higher magnitude compared to those for interband transitions, which are located in the near-infrared range. The increase in the magnetic field results in a blue-shift of the 2PA spectra for both intraband and interband transitions, whereas changes in the azimuthal quantum number only affect the spectra for interband transitions.

Investigation of two-photon 2s→1s decay in one-electron and one-muon ions

We study the radiative decay of the $2s$ state of one-electron and one-muon ions, where the two-photon mechanism plays an important role. Due to the nuclear size corrections, the radiative decay of the $2s$ state in the electron and muon ions is qualitatively different. Based on the accurate relativistic calculation, we introduce a two-parameter approximation, which makes it possible to describe the two-photon angular-differential transition probability for the polarized emitted photons with high accuracy. The emission of photons with linear and circular polarizations is studied separately. We also investigate the transition probabilities for the polarized initial and final states. The investigation is performed for ions with atomic numbers $1\ensuremath{\le}Z\ensuremath{\le}120$.

Preparation, structure and optical properties of mixed stacked cocrystals of 2,2':6′,2″-Terpyridine and 1,2,4,5-tetracyanobenzene

Organic charge-transfer cocrystals exhibit fantastic physical properties owning to their multi-component synergistic and collective effect. In spite of the abundant material supplies, the cocrystals people prepared so far are far less than expectation. Here, we reported the preparation, structure and optical properties of a mixed stacked cocrystal of 2,2':6′,2″-terpyridine and 1,2,4,5-tetracyanobenzene. The structural and optical properties were analyzed by using single-crystal X-ray diffraction (SCXRD), FTIR, Raman, optical absorption and emission spectra experimentally and the molecular electrostatic potential (MEP), Hirshfeld surface, the reduced density gradient (RDG), frontier molecular orbitals (FMO) theory, inter-fragment charge transfer (IFCT) analysis and time dependent density functional theory (TD-DFT) theoretically. The TTC sample exhibits strong up-conversion luminescence that is absent in their individual components. Theoretical analysis indicates that the NLO properties of TTC may be due to a significant increase in two-photon transition probabilities δTPA related to intermolecular charge transfer (CT) interactions. Our results may be helpful for a better understanding of NLO mechanism in ground state CT cocrystals.

Time dependent variation perturbation calculation of two-photon transition probability and hyperfine shift in hydrogen atom under plasma environment

A new theoretical approach has been adopted to calculate the non-relativistic parity conserved two-photon transition probability (TPTP) from ground to excited S and D states along with the hyperfine splitting (HFS) of ground and excited S states of H atom within principal quantum number n≤4 under classical and quantum plasma environment represented by screened potential model. The methodology involves direct calculation of TPTP via an intermediate virtual state obtained from time dependent variation perturbation formalism. The standard formula with sum over states representing the TPTP has been replaced by its variational equivalent obtained from currently established procedure for two-photon excitation via fourth order time dependent variation perturbation theory. Unlike previous methodologies the current formalism yields a compact representation of the physical nature of the intermediate virtual state having well defined profile. The HFS has been evaluated under first order perturbation theory. The plasma embedded ground state energy is obtained from standard diagonalization procedure. Finite Slater type basis sets have been used for the ground and all the excited states. The variational coefficients for the excited states are determined from optimization of an appropriate time averaged variational functional. Results for free H atom agree well with existing estimates for all the states. The results for the plasma embedded states follow smooth and interesting pattern.

Polarization dependence of 133Cs 6S1/2-6P3/2-11S1/2 electromagnetically induced transparency at room temperature.

The effect of polarization on the ladder-type electromagnetically induced transparency (EIT) spectra of 133Cs atoms at room temperature for the transitions 62P1/2-62P3/2-112S1/2 is experimentally studied. The entire spectra with additional peaks arising from the Doppler effect are observed. As the relative angle between the probe's and coupling's plane of polarization arranges at 0°, 45°, and 90°, the peak height ratio of 44'3" to 44'4" increases by more than 7 times with corresponding values of 0.19, 0.69, and 1.4. Meanwhile, that of 45'4" to 44'4" are found to be 0.61, 0.87, and 1.23 (doubled), respectively. A theoretical model built to explain the experimental results with the considerations of optical pumping effect, two-photon transition probability, dephasing rate, and integration all over the velocity distribution. The simulation and experimental results are well-agreed.

Diffraction and quantum control of wave functions in nonresonant two-photon absorption

In this study, the nonresonant two-photon absorption process in a two-level atom, induced by a weak chirped pulse, is theoretically investigated in the frequency domain. An analytical expression of the wave function expressed by Fresnel functions is obtained, and the two-photon transition probability (TPTP) versus the integral bandwidth, spectral width, and chirp parameter is analyzed. The results indicate that the oscillation evolution of the TPTP result from quantum diffraction of the wave function, which can be explained by analogy with Fresnel diffraction from a wide slit in the spatial domain. Moreover, the ratio between the real and imaginary parts of the excited state wave function and, hence, the atomic polarization, can be controlled by the initial phase of the excitation pulse. In some special initial phase of the excitation pulse, the wave functions with purely real or imaginary parts can be obtained by measuring the population probability. This work provides a novel perspective for understanding the physical details of the interactions between atoms and chirped light pulses in the multiphoton process.

A Floquet formalism for the interaction of magnetically trapped atoms with rf fields

A many-mode Floquet theory (MMFT) formalism is applied to study the interaction of a polychromatic rf field with cold atoms trapped in a quadrupole magnetic trap. The validity of the MMFT approach is first established by comparing its results with those of the previously used formalisms for the cases of single- and two-frequency rf fields. The effect of truncation of the infinite matrices has been studied and a suitable truncation order is obtained for the numerical implementations. The results obtained for the composite atom–field system has shown some exquisite features such as lattice-like periodic variation in the eigen-energies and large two-photon transition probabilities between the atomic states. Dependence of the trapping parameters such as trap position, depth and width on the rf field parameters like field strength and mode separation has also been studied. This work predicts the generation of lattice type atom-trapping geometries which can be controlled by varying the rf field parameters of a polychromatic rf field.

A Theoretical Study of One- and Two-Photon Activity of D-Luciferin

In the present work, we have theoretically studied the one and two-photon absorption (OPA and TPA) probabilities of the native D-luciferin molecule and attempted to find the origin of its larger TPA cross-sections in polar solvents than in non-polar ones. The calculations using state-of-the-art linear and quadratic response theory in the framework of time-dependent density functional theory using hybrid B3LYP functional and cc-pVDZ basis set suggests that two-photon transition probability of this molecule increases with increasing solvent polarity. In order to explicate our present findings, we employed the generalized few-state-model and inspected the role of different optical channels related to the TPA process. We have found that the two-photon transition probability is always guided by a destructive interference term, the magnitude of which decreases with increasing solvent polarity. Furthermore, we have evaluated OPA parameters of D-luciferin and noticed that the the excitation energy is in very good agreement with the available experimental results.

Real-time tracking of two-electron dynamics in the ionization continuum of Xe

Real-time tracking of atomic two-electron coherent dynamics is investigated through excitation of autoionizing wave packets in the ionization continuum of xenon. Extreme-ultraviolet high-order harmonics of ultrashort Ti:Sapphire laser pulses start the wave packets in energy ranges with a dense covering by two-valence-electron and single-inner-valence-electron excited-state resonances. The 15th harmonic covers resonances of type $5s5{p}^{6}nl$ with principle quantum number $n$ ranging from $n\ensuremath{\approx}14$ up to $n\ensuremath{\approx}21$ and one resonance of type $5{s}^{2}5{p}^{4}nl{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ with fixed $n,\phantom{\rule{0.16em}{0ex}}{n}^{\ensuremath{'}}$ and orbital angular momenta $l,\phantom{\rule{0.16em}{0ex}}{l}^{\ensuremath{'}}$. In parallel the 17th harmonic starts a wave packet beyond the Xe $5s5{p}^{6}$ ionization threshold where exclusively resonances contribute with two-valence electrons being excited. The evolution in time of these wave packets is probed by inducing a continuum-continuum transition with the Ti:Sapphire laser pulses at variable delay times and detecting the photoelectrons after this two-photon transition. The dependence of the experimental data on the pump-probe delay time can be approximated by a two-photon transition probability derived on the basis of Fano's theory.

Donors contribute more than acceptors to increase the two-photon activity--a case study with cyclopenta[b]naphthalene based molecules.

In the present work, we address the question -"which among the electron donors and the electron acceptors contribute more to the two-photon (TP) activity of a donor-π-acceptor type of molecule?" For this purpose we have performed ab initio calculations to calculate the TP transition probability (δTP) of a recently synthesized (Benedetti et al., J. Am. Chem. Soc., 2012, 134(30), 12418-12421) cyclopenta[b]naphthalene based chemo-sensor and its derivatives containing different electron donor and acceptor groups. Our study revealed that both under vacuum and in solvent phases, an increase in electron donor strength (-OMe, -NH2, -NMe2) increases the δTP value up to five times, whereas, an increase in the acceptor group strength (-COCH3, -NO2, -CN) increases it by a factor of two only. The highest δTP value is obtained for the molecule having the strongest donor-acceptor pair (-CN, -NMe2) considered in this work. We have also noted that, the removal of the cyclopentane ring from the original system increases the δTP value by ∼20% and the replacement of the naphthyl group by the benzene ring decreases it by ∼70%. All these results are explained by inspecting different TP tensor elements and different transition moment vectors involved in a two-state model approach. A close scrutiny of different parameters in 2SM clearly reveals that upon increasing the strength of either the donor or the acceptor group the parameters change in favour of increasing the overall δTP values but in the case of donors this effect is much larger.

Chemical control of channel interference in two-photon absorption processes.

The two-photon absorption (TPA) process is the simplest and hence the most studied nonlinear optical phenomenon, and various aspects of this process have been explored in the past few decades, experimentally as well as theoretically. Previous investigations have shown that the two-photon (TP) activity of a molecular system can be tuned, and at present, performance-tailored TP active materials are easy to develop by monitoring factors such as length of conjugation, dimensionality of charge-transfer network, strength of donor-acceptor groups, polarity of solvents, self-aggregation, H-bonding, and micellar encapsulation to mention but a few. One of the most intriguing phenomena affecting the TP activity of a molecule is channel interference. The phrase "channel interference" implies that if the TP transition from one electronic state to another involves more than one optical pathway or channel, characterized by the corresponding transition dipole moment (TDM) vectors, the channels may interfere with each other depending upon the angles between the TDM vectors and hence can either increase (constructive interference) or decrease (destructive interference) the overall TP activity of a system to a significant extent. This phenomenon was first pointed out by Cronstrand, Luo, and Ågren [Chem. Phys. Lett. 2002, 352, 262-269] in two-dimensional systems (i.e., only involving two components of the transition moment vectors). For three-dimensional molecules, an extended version of this idea was required. In order to fill this gap, we developed a generalized model for describing and exploring channel interference, valid for systems of any dimensionality. We have in particular applied it to through-bond (TB) and through-space (TS) charge-transfer systems both in gas phase and in solvents with different polarities. In this Account, we will, in addition to briefly describing the concept of channel interference, discuss two key findings of our recent work: (1) how to control the channel interference by chemical means, and (2) the role of channel interference in the anomalous solvent dependence of certain TP chromophores. For example, we will show that simple structurally induced changes in certain dihedral angles of the well-known betaine dye (TB type) will help fine-tune the constructive channel interference and hence increase the overall TP activity of molecules with this general TP channel structure. Another intriguing result we will discuss is observed for a tweezer-trinitrofluorinone complex (TS type) where, on moving from polar to essentially nonpolar solvents, the nature of the channel interference switches from destructive to constructive, leading to a net abnormal solvent dependence of the TP activity of the system. The present Account highlights the usefulness of the channel interference effect and establishes it as a new and unique way of controlling the TP transition probability in different types of three-dimensional molecules.

Quantum focusing and coherent control of nonresonant two-photon absorption in frequency domain.

We theoretically investigate the nonresonant two-photon absorption (TPA) process in a two-level atom induced by a weak chirped pulse in the frequency domain. According to the extremum condition of the two-photon transition probability (TPTP) at the transition center frequency, we propose a Fresnel-inspired pulse tailoring scheme for TPA that is significantly different from that of Broers etal. [Phys. Rev. A46, 2749 (1992)]. Using this scheme, the TPTP can be focused or eliminated completely by constructively or destructively modulating various pathways of the quantum interference. Our results are a significant improvement on those obtained by Broers etal. and will have potential applications in selective two-photon microscopy and spectroscopy.

Quantum interference control of a four-level diamond-configuration quantum system

We investigate coherent control of the two-photon transition pathways of a four-level atomic system in a diamond configuration. When an ultrashort laser pulse interacts with this system in the ground state 5S1/2 of rubidium, the two-photon transition probability amplitude of 5D3/2 is obtained by a summation of all possible resonant and nonresonant two-photon transition probability amplitudes via 5P1/2 and 5P3/2. Second-order perturbation theory predicts that the maximal constructive interference of the transition probability amplitudes occurs when the phases of eight different spectrum blocks satisfy four different phase relations. Experiments carried out with spectrally phase-coded laser pulses show good agreement with the theoretical prediction.

Manipulation of Resonance-Enhanced Multiphoton-Ionization Photoelectron Spectroscopy by Two Time-Delayed Femtosecond Laser Pulses

We theoretically demonstrate that the (2+1) resonance-enhanced multiphoton-ionization (REMPI) photoelectron spectrum in a cesium (Cs) atom can be effectively manipulated by two time-delayed femtosecond laser pulses, involving its photoelectron spectral structure and photoelectron energy. We show that the photoelectron spectrum exhibits interference fringes and the fringe spacing is determined by the time delay of the two laser pulses, and the photoelectron energy is periodically modulated and the modulation period is determined by the two-photon transition frequency of the excited state. Finally, we utilize the power spectrum of the two time-delayed laser pulses and the two-photon transition probability of the excited state to respectively explain the modulations of the photoelectron spectrum and photoelectron energy.

Two-photon decay of inner-shell vacancies in heavy atoms

Based on the second-order perturbation theory, we investigate the two-photon decay of $K$-shell vacancies in heavy atoms. The many-electron transition amplitude that occurs in the theory is evaluated by means of the independent particle approximation (IPA). By using this approach, computations are performed for the decay of neutral gold and are directly compared with recent experimental data, not relying on any scaling assumptions. The obtained results confirm previously identified discrepancies between the IPA theory and the experiment for the $2s\ensuremath{\rightarrow}1s$ transition, and an apparent ``resonance'' region of the $3s\ensuremath{\rightarrow}1s$ transition, but they show a moderate agreement with the measured data for the $3d\ensuremath{\rightarrow}1s$ and $4s+4d\ensuremath{\rightarrow}1s$ cases. Moreover, with the help of the IPA we discuss the validity of the nonrelativistic scaling that was employed in the past to estimate the relative two-photon transition probabilities $P$ in heavy atoms based on calculations done for lighter elements and different decay geometries. We find, in particular, that the electric-dipole angular distribution of emitted photons holds rather well even in the high-$Z$ domain, while the assumption that the relative probability $P$ is independent of nuclear charge may result in 10--30% inaccuracy of theoretical predictions.

NON-RESONANT TWO-PHOTON ABSORPTION CONTROL BY TWO TIME-DELAYED LASER PULSES

In this paper, we theoretically investigate the control of the non-resonant two-photon absorption induced by two time-delayed laser pulses, and an analytical solution for the dependence of the two-photon transition probability on the time delay and relative carrier-envelope phase of the two laser pulses is achieved. We show that the two-photon absorption can be significantly enhanced or completely suppressed by varying the time delay between the two laser pulses or their relative carrier-envelope phase. We also show that the two-photon absorption can be selectively excited when two excited states are simultaneously excited. Furthermore, we discuss the two-photon absorption control in the molecular system and analyze the effect of the absorption bandwidth on the control efficiency of the two-photon absorption.

Polarization and phase control of two-photon absorption in an isotropic molecular system

We theoretically and experimentally study the polarization and phase control of two-photon absorption in an isotropic molecular system. We theoretically show that the two-photon transition probability decreases when the laser polarization changes from linear through elliptical to circular, and the laser polarization does not affect the control efficiency of two-photon transition probability by shaping the spectral phase. These theoretical results are experimentally confirmed in coumarin 480. Furthermore, we propose that the combination of the laser polarization with the spectral phase modulation can further increase the control efficiency of the two-photon absorption.

Enhancement of Twist Angle Dependent Two-Photon Activity through the Proper Alignment of Ground to Excited State and Excited State Dipole Moment Vectors

Herein, we show that the two-photon (TP) transition probability (δTP) of o-betaine system will reach its maximum value at a twist angle around 65°. However, the potential energy scan with respect to the twist angle between its two rings indicates that the molecule in its ground state is quite unstable at this twist angle. Out of the different possibilities, the one having a single methyl group at the ortho position of the pyridinium ring is found to attain the optimum twist angle between the two rings, and interestingly, this particular substituted o-betaine has larger δTP value than any other substituted or pristine o-betaine. The twist angle dependent variation of δTP has been explained by employing the generalized-few-state-model formula for 3D molecules. The results clearly reveal that the magnitude of ground to excited state and excited state dipole moment vectors as well as the angle between them are strongly in favor of maximizing the overall δTP values at the optimum twist angle. The constructive interference between the optical channels at the optimum twist angle also plays an important role to achieve the maximum δTP value. Furthermore, to give proper judgment on our findings, we have also performed solvent phase calculations on all the model systems in nonpolar solvents, namely, cyclohexane and n-hexane, and the results are quite consistent with the gas phase findings. The present study will definitely offer a new way to synthesize novel two-photon active material based on o-betaine.

High-Polarity Solvents Decreasing the Two-Photon Transition Probability of Through-Space Charge-Transfer Systems - A Surprising In Silico Observation.

In the Letter, we address the question as to why larger two-photon absorption cross sections are observed in nonpolar than in polar solvents for through-space charge-transfer (TSCT) systems such as [2,2]-paracyclophane derivatives. In order to answer this question, we have performed ab initio calculations on two well-known TSCT systems, namely, a [2.2]-paracyclophane derivative and a molecular tweezer-trinitrofluorinone complex, and found that the two-photon transition probability values of these systems decreases with increasing solvent polarity. To rationalize this result, we have analyzed the role of different optical channels associated with the two-photon process and noticed that, in TSCTs, the interference between the optical channels is mostly destructive and that its magnitude increases with increasing solvent polarity. Moreover, it is also found that a destructive interference may sometimes even become a constructive one in a nonpolar solvent, making the two-photon activity of TSCTs in polar solvents less than that in nonpolar solvents.

Polarization dependence of double-resonance optical pumping and electromagnetically induced transparency in the5S1/2−5P3/2−5D5/2transition of87Rb atoms

The polarization dependence of double-resonance optical pumping (DROP) in the ladder-type electromagnetically induced transparency (EIT) of the $5{S}_{1/2}\ensuremath{-}5{P}_{3/2}\ensuremath{-}5{D}_{5/2}$ transition of ${}^{87}$Rb atoms is studied. The transmittance spectra in the $5{S}_{1/2}(F=2)\ensuremath{-}5{P}_{3/2}({F}^{\ensuremath{'}}=3)\ensuremath{-}5{D}_{5/2}({F}^{\ensuremath{'}\ensuremath{'}}=2,3,4)$ transition were observed as caused by EIT, DROP, and saturation effects in the various polarization combinations between the probe and coupling lasers. The features of the double-structure transmittance spectra in the $5{S}_{1/2}(F=2)\ensuremath{-}5{P}_{3/2}({F}^{\ensuremath{'}}=3)\ensuremath{-}5{D}_{5/2}({F}^{\ensuremath{'}\ensuremath{'}}=4)$ cycling transition were attributed to the difference in saturation effect according to the transition routes between the Zeeman sublevels and the EIT according to the two-photon transition probability.