An analytical study on absorptive lineshape of a driven N-type open four-level system: Quantum interference effects

An open four-level system of having two pairs of closely spaced levels (N-type configuration) is driven by a single electromagnetic field and tuned resonant with the average frequency of four dipole allowed transitions. Under the Doppler free condition and by using a semiclassical formulation of atom-field interaction for four dipole allowed transitions, we derive the optical Bloch equations for the said four-level system coupled to the driving field. In order to obtain the field induced polarization and hence the absorptive lineshapes, we use the usual perturbation method for getting the approximate analytical solution to the coupled optical Bloch equations for the density matrix elements. Through the off-diagonal complex density matrix elements, we introduce the field dependent phase angles arising out of the quantum interference between the levels participating in dipole allowed transitions. The difference between the field dependent and field independent phases are pointed out. In particular, we investigate the effects of Rabi frequencies and the field dependent phases on the absorptive lineshape. The analytical expressions for the effective linewidths, effective detunings and the induced polarization clearly indicate the role of quantum interference.     

A theoretical analysis on coherent double resonant absorptive lineshape in closely spaced transitions for λ − type five level system

Under Doppler free double resonant condition, an open five level system with three closely spaced upper levels and a pair of closely spaced lower levels, is allowed to interact with two electromagnetic fields. In the domain of semiclassical formulation of atom-field interaction, the optical Bloch equations involving the density matrix elements are constructed. These coupled optical Bloch equations are unsolvable in closed analytical form. We use the perturbation method for getting the approximate analytical solutions to the coupled optical Bloch equations. The absorptive signal lineshape corresponding to pump frequency Ω₁ and signal frequency Ω₂ is obtained. We also obtain the absorptive signal line shape by interchanging the pump and signal (i.e., the pump frequency is Ω₂ and the signal frequency is Ω₁) mutually. The interferences between the probability amplitudes for different energy levels involving the dipole allowed and the dipole forbidden transitions give rise to the field dependent and field independent quantum coherence respectively. With the suitable manipulation of the coherence between the two lower levels, the signal lineshapes for on-resonance and off-resonance pump positions are explored in a great detail. The off-resonance pump leads to the two-photon absorption and hence the signature of the nonlinear resonances. On the other hand, the on-resonance pump positions lead to the Rabi splitting. The shifts of the resonance peak positions are explored as a function of pump intensity and the level spacings of the closely spaced levels.     

Effects of spontaneous decay on absorptive lineshape in closely spaced transitions of an open four level system

A semi classical treatment of atom field interaction is adopted under Doppler free condition, for an open four level system with three closely spaced upper levels. The field induced polarization in closed analytical form is used in order to examine the absorption lineshape. Through the off-diagonal density matrix elements, the introduction of phase angles between the levels participating in dipole allowed / forbidden transitions are automatic. By the proper control of ground level decay and field independent coherence, we obtain inversionless gain.     

Variance squeezing and information entropy squeezing via Bloch coherent states in two-level nonlinear spin models

The nonclassical squeezing effect emerging from a nonlinear coupling model (generalized Jaynes–Cummings model) of a two-level atom interacting resonantly with a bimodal cavity field via two-photon transitions is investigated in the rotating wave approximation. Various Bloch coherent initial states (rotated states) for the atomic system are assumed, i.e., (i) ground state, (ii) excited state, and (iii) linear superposition of both states. Initially, the atomic system and the field are in a disentangled state, where the field modes are in Glauber coherent states via Poisson distribution. The model is numerically tested against simulations of time evolution of the based Heisenberg uncertainty relation variance and Shannon information entropy squeezing factors. The quantum state purity is computed for the three possible initial states and used as a criterion to get information about the entanglement of the components of the system. Analytical expression of the total density operator matrix elements at t > 0 shows, in fact, the present nonlinear model to be strongly entangled, where each of the definite initial Bloch coherent states is reduced to statistical mixtures. Thus, the present model does not preserve the modulus of the Bloch vector.     

Modification of the Robert-Bonamy formalism in calculating Lorentzian half-widths and shifts

We have found an invalid assumption in the Robert-Bonamy formalism that has been widely used for calculating Lorentzian spectral line half-widths and shifts for decades. The problem results in their derivation where they assumed the cumulant expansion can be used to evaluate the Liouville matrix element . At first sight, their assumption appears to be correct because this matrix element is diagonal in the Liouville space and as a result, it looks like that a basic requirement in applying the cumulant expansion is satisfied. However, by decomposing it into two Hilbert matrix elements associated with S_I and S_F, respectively, we have found that neither of these is diagonal in Hilbert space. Therefore, their assumption is not valid and their expressions for the half-widths and shifts are incorrect. We have found by choosing an average over the internal degrees of the bath molecule as the average in the cumulant expansion, one is able to apply this expansion properly and obtain the correct expressions. Numerical calculations show new half-width and shift values differ from previous ones, and the stronger the interaction between two molecules is, the larger these differences are.     

Spatial evolutions of inversionless gain and lasing field intensity in an open V-type system with SGC

This paper studies manly spatial evolution of gain without inversion (GWI) and the Rabi frequency E (intensity ?_p) of the probe field in an open V-type three-level inversionless lasing system with spontaneously generated coherence (SGC) for both cases with and without Doppler broadening. We found that: (1) Varying sizes of SGC strength (measured by angle θ), atomic exit rate (r₀) and ratio (S) of the atomic injection rates has remarkable effect on spatial evolutions of GWI and E (?_p). This effect in the case with Doppler broadening is similar to but weaker than that in the case without Doppler broadening. (2) Regardless of that Doppler broadening is present or not, GWI and E (?_p) increase with increase of θ, r₀ and S in certain value ranges of θ, r₀ and S; in the case with SGC we can obtain GWI and E (?_p) much larger than those in the case without SGC, while by choosing values of γ₀ and S, in the open system we can obtain LWI gain and E (?_p) much larger than those in the corresponding closed system. (3) The propagation distance in which GWI exists in the case with Doppler broadening is longer than that in the case without Doppler broadening; in the case without Doppler broadening, we can obtain larger GWI than that in the case with Doppler broadening; but in the case with Doppler broadening, we can obtain larger E (?_p) than that in the case without Doppler broadening.     

High resolution Raman spectroscopy of the fundamental vibrational band of 16O2

The vibrational Raman spectrum of ¹⁶O₂ has been recorded with high resolution (0.05 cm^?1 for the Q branch). The expansion of the Hamiltonian as a sum of irreducible tensors of the O(3) group allowed us to obtain easily the expressions for the energy levels, taking into account the off-diagonal matrix elements. From the analysis of the spectrum the excited state constants have been calculated; in particular the rotational constants obtained are: B₁ = 1.421884 ± 0.000013 cm^?1 and D₁ = (?4.864 ± 0.014)10^?6 cm^?1.     

Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

The residual interaction of bound nucleons-two-nucleon matrix elements deduced from transfer experiments

Matrix elements for the effective two-nucleon interaction have been deduced from the population of multiplets near closed shells as observed in direct transfer reactions. In the evaluation, the limited purity of such multiples was taken into consideration, typically by weighting the observed fractions of the two-nucleon configurations by their spectroscopic strenghts and by using the resulting energy centroids. In a few cases, off-diagonal matrix elements are available from empirical wave funcitons. The systematic errors for particle-particle matrix elements extracted directly and those obtained from Pandya transformations were found to go in opposite directions. In some cases, this feautre of the empirical mehtod could be used to suggest upper and lower “bounds” for the extracted matrix elements. Diagonal matrix elements for the empirical residual interaction show a number of features suggestive of an underlying simplicity in the interaction of bound nucleons. Within experimental uncertainties (of about 10% for T=0 matrix elements) the monopole parts of the matrix elements are fit well with a simple A?0.75 dependence, and the data available to date do not reveal any significant monopole dependence on the quantum numbers of the interacting nucleons. The usefulness of scaling is suggested. Generally, diagonal matrix elements E_J(j₁, j₂) normalized by the extracted A-dependent monopole strength agree within expected experimental uncertainties whether derived from particle-particle or particle-hole multiples and whether extracted from the beginning or the end of a major shell. For values J≠0, the diagonal E_J(j²) matrix elements seem to follow two universal functions which depend on the semi-classical coupling angles θ₁₂, but are otherwise independent on j. For j₁≠j₂ several “typical” functions ?(θ₁₂) can be constructed which fit subsets of the data and differ in a predictable way. The general features of the bound-nucleon interaction appear consistent with those of theoretical matrix elements based on a number of short-range model forces or on calculations using the G matrix approach to deal with realistic free nucleon forces. For the latter, the available theoretical numbers for j₁=j₂ agree well with the T=1 set, but they differ quantitatively from the observed matrix elements for T=0, sometimes by many (experimental) standard deviations.     

Constraints for anomalous dimensions of local light-cone operators in [ϕ 3]6 theory

Using the MS scheme, we derive in [? ³]₆ theory the collinear conformal Ward identity for the Green's functions of local light-cone operators of leading twist. The Ward identity for special collinear conformal transformations and renormalization group invariance give constraints for the off-diagonal part of the anomalous dimension matrix for the general case of β#0. We compute the anomaly of special conformal transformation in lowest loop order and obtain from the constraints the off-diagonal part of the anomalous dimension in 2-loop order.     

VPT2+K spectroscopic constants and matrix elements of the transformed vibrational Hamiltonian of a polyatomic molecule with resonances using Van Vleck perturbation theory

Vibrational levels of polyatomic molecules are analysed with Van Vleck perturbation theory to connect experimental energy levels to computed molecular potential energy surfaces. Vibrational matrix elements are calculated from a quartic potential function via second-order Van Vleck perturbation theory, a procedure that treats both weak and strong interactions among vibrational states by approximately block-diagonalising the vibrational Hamiltonian. A clear and complete derivation of anharmonic and resonance constants as well as general expressions for both on- and off-diagonal matrix elements of the transformed Hamiltonian is presented. The equations are written in partial fraction form and as a constant multiplied by a harmonic oscillator matrix element to facilitate removing the effect of strongly interacting resonant states both in analytical formulae and in computer code. The derived equations are validated numerically, and results for the isotopomers of formaldehyde (H₂CO, HDCO, D₂CO) are included. The implications of the equations on zero-point energy calculations and experimental fits are discussed. The VPT2+K method is defined by these results for use in fitting and calculating vibrational energy levels.     

Nuclear matrix elements in the 7/2−–9/2+ beta decay of175Yb

The angular correlation of the \({7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0em} 2}^ - \xrightarrow[{353keV}]{\beta }{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-0em} 2}^ + \xrightarrow[{114keV}]{\gamma }{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0em} 2}^ + \) is measured atE _β =300 keV and found to beA ₂(β,γ)=0.108 ± 0.028. The result is combined with the data on longitudinal polarisation nuclear orientation to obtain the nuclear matrix elements which would give a simultaneous fit to all experimental data. The matrix elements thus obtained have predicted theβ spectrum shape correction factor. The sizes of the matrix elements indicate a cancellation effect in vector matrix elements which explains deviation from ξ-approximation. From the ratio of higher order matrix elements,λ, the deviation from theCVC ratio due to Fujita is found and thus the importance of off-diagonal elements inH _c is noted. The experimental matrix elements are compared with the model-predicted ones.     

The dependence of the photodisintegration cross section on the off-shell T-matrix

Nucleon-nucleon scattering phase shifts determine the diagonal element of the transition matrix. The off-diagonal elements are not completely arbitrary but have conditions imposed on them by the range and the tail of the potential. Electromagnetic interaction can also be used to place restrictions on the off-diagonal elements. We find that the cross section of the deuteron photodisintegration is sensitive to the off-shell transition matrix. The integrated cross section can be varied by as much as 30 % or more, and the matrix element for the El transition by a factor of 2. While the matrix element for the photodisintegration depends on the off-shell elements of the T-matrix, it cannot be used to discriminate between alternative off-shell T-matrices. We have constructed classes of different off-shell T-matrices, which produce identical photo-disintegration cross sections and other two-body scattering and bound-state properties.     

On generalized Waldmann-Snider equations: II. Π̌-subdynamics

Three non-equilibrium equations have been derived describing the evolution of the matrix elements, which are highly off-diagonal on the internal energy of the singlet density operator, ?⁽¹⁾, of a quantum dilute gas of molecules with internal levels. The first equation has been obtained by assuming the requirement of sufficient spacing of the spectral transition lines, the second one being its extension for weakly inhomogeneous gases and the third one the equation generalized for spectral transition lines of arbitrary spacing. They have been obtained inside the framework of the subdynamics theory of the Brussels Group where the Π̌-subdynamics have been explicitly constructed.Their linearized version is presented permitting to obtain closed equations for the ?⁽¹⁾-operator, which can be shown useful in the correlation functions, being interesting in spectroscopic measurements. A discussion concerning the above aspects has been included as well.     

Collisional line shifting and broadening in the fundamental P-branch of CO in Ar between 214 and 324 K

The collisional shifts and widths of several P-branch spectral lines in the fundamental band of CO-Ar have been measured at temperatures between 214 and 324 K and pressures between 0.025 and 1 atm. The widths have been determined using a line shape model based on the solution of the transport/relaxation equation for the appropriate off-diagonal element of the density matrix. The model uses a realistic molecular potential energy surface to calculate the speed dependence of the collisional broadening, and a rigid sphere potential to calculate the translational motion. It is found that both the shifting and broadening coefficients follow a power law dependence on the temperature. Additionally, it is demonstrated that studies have tended to overestimate the accuracy of collisional widths when the line shape model used to obtain the widths involves multiple fitted line shape parameters or fails to fit the measured spectra within the experimental noise.     

Entanglement and Berry Phase in a Parameterized Three-Qubit System

In this paper, we construct a parameterized form of unitary \(\breve {R}_{123}(\theta _{1},\theta _{2},\varphi )\) matrix through the Yang-Baxterization method. Acting such matrix on three-qubit natural basis as a quantum gate, we can obtain a set of entangled states, which possess the same entanglement value depending on the parameters ?? ₁ and ?? ₂. Particularly, such entangled states can produce a set of maximally entangled bases Greenberger-Horne-Zeilinger (GHZ) states with respect to ?? ₁ = ?? ₂ = π/2. Choosing a useful Hamiltonian, one can study the evolution of the eigenstates and investigate the result of Berry phase. It is not difficult to find that the Berry phase for this new three-qubit system consistent with the solid angle on the Bloch sphere.     

Application of microwave amplitude modulation technique to measure spin-lattice and spin-spin relaxation times accurately with a continuous-wave EPR spectrometer: Solution of Bloch’s equations by a matrix technique and least-squares fitting

A matrix technique to calculate signals recorded using the microwave amplitude-modulation technique is described. The calculations are carried out for spin packets, on and off resonance, to take into account inhomogeneous broadening. Both, the transverse component of magnetization representing the continuous-wave signal in a resonator, such as a cross-looped resonator, as well as the signal (electromotive force) induced in a pickup coil oriented parallel to the external magnetic field, are calculated for an arbitrary value of the coefficient of modulation. This is accomplished by solving the relevant Bloch equations in the rotating frame for the case when the amplitude of the microwave field is modulated by a sinusoidal wave, using Fourier expansions of the longitudinal and transverse components of the magnetization in Bloch equations. This results in a series of coupled equations inM _α(n) (α=y,zz), the magnetic moments of vaarious orders, leading to a penta-diagonal matrix of infinite dimension. These equations are then truncated and solved by a fast matrix technique to calculateM _α(n), required to calculate the modulation signals as functions of the amplitudemodulation frequency Ω. It is outlined how to exploit the expressions for the modulation signals to estimate the spin-lattice relaxation timesT ₁ and spin-spin relaxation timesT ₂ accurately by the leastsquares procedure, fitting simultaneously all signals obtained for spin packets, on and off resonance, at various modulation frequencies. Illustrative examples are provided.     

Effect of spontaneously generated coherence on Kerr nonlinearity in a four-level atomic system

We propose a scheme for giant enhancement of the Kerr nonlinearity in a four-level atomic system in which spontaneously generated coherence is present. The physics mechanism of the enhancement of Kerr nonlinearity is mainly based on the presence of an extra atomic coherence induced by the spontaneously generated coherence. Numerical values obtained by solving the density matrix equations agree well with these exact analytical values.     

Dynamics of collective decoherence and thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part.We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction is of order N², while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins.Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations.