Time-dependent Density-matrix Equations Research Articles

Spectral features of electromagnetically induced absorption in 85Rb atoms

We present theoretical and experimental studies of electromagnetically induced absorption (EIA) for the transition of 85Rb-D2 line. From the results calculated by solving time-dependent density-matrix equations including the optical and Zeeman coherences connected via three-photon interactions, the EIA signals have ultra-narrow feature in low powers of coupling beam in both same- and orthogonal-polarization configurations for both stationary and thermal atoms. However, the ultra-narrow EIA signals from high powers of coupling laser beam still remained in the ultra-narrow state only in the same-polarization configuration for thermal atoms. The calculated results match well with experimental results except ultra-narrow EIA region considering linewidths of two separate coupling and probe lasers.

Density matrix equation analysis of optical–optical double-resonance multiphoton ionization probability

An analytical formula of the optical–optical double-resonance multi-photon ionization (OODR-MPI) probability is derived from the time-dependent density-matrix equations that describe the interaction of photon and material. Based on the formula, the variation of the multiphoton ionization (MPI) probability with laser resonance detuning, Rabi frequency, laser pulse duration and ionization rate is investigated theoretically. It is shown that the MPI probability will decrease with the increase of laser resonance detuning, to some extent, to zero. The influence of the pump laser resonance detuning on the ionization probability is more important with respect to the probe laser. It not only influences Rabi frequency for saturation, but also the saturation value of MPI probability. The MPI probability will increase with Rabi frequency, laser pulse duration and ionization rate. It is also found that though the variation of the populations in the ground, the first and the second resonance states is different at the beginning of laser radiation, but they will still decrease to zero as the time goes on. It is then that the ionization probability gets the maximum value. Thus long laser pulse duration and high laser intensity are in favor for improving the MPI probability. These theoretical research results can provide a useful guide for the practical application of OODR-MPI spectroscopy. • An analytical expression of OODR-MPI probability has been derived. • MPI probability decreases with the increase of laser resonance detuning. • The influence of pump laser on the MPI probability is larger than probe laser. • Larger laser pulse duration and intensity are in favor of higher MPI probability.

Analytical approach to dynamic cancelation of the AC Stark shift in a pulsed three-level optical lattice clock

We study analytically the dynamic cancelation of the AC Stark shift in a three-level pulsed optical lattice clock by solving the time-dependent density-matrix equations in the adiabatic following limit. We found an optimum value of the common detuning Δ0 of two coherent driving fields at which the dynamic AC Stark shift cancelled completely at an arbitrary field strength. The optimum value of Δ0 for the well-studied Sr atoms agreed well with the values obtained by using other methods, for instance, a numerical calculation and a wave-function formalism. We give an optimum value of common detuning, Δ0 = 65.3 MHz, for the pulsed Raman Yb optical lattice clock.

Observation of query pulse length dependent Ramsey interference in rubidium vapor using pulsed Raman excitation

We report observation of query pulse length dependent Ramsey interference (QPLD-RI), using pulsed Raman excitation in rubidium vapor. This is observed when a long, attenuated query pulse is used during pulsed Raman excitation. We explain the physical mechanism behind the QPLD-RI using a Bloch vector model. We also use numerical solutions to time-dependent density matrix equations to simulate this interference effect, showing qualitative agreement with experimental results. Presence of such interference could create a potential source of error in a vapor cell Raman clock constructed using frequency-domain Ramsey interference (FDRI).

Numerical simulation of the resonance photoionization probability by density-matrix equation

An analytical expression of the resonance photoionization (PI) probability is deduced from the time-dependent density-matrix equations that describe the interaction of laser field and material. Based on this expression, the variation of PI probability as a function of some parameters such as laser resonance detuning and Rabi frequency are studied theoretically. It is shown that the increase of the laser resonance detuning will make the PI probability decrease, to some extent, to zero. Both the Rabi frequency and the laser pulse duration will also make the PI probability increase to saturation. It is also found that the variation of PI probability with Rabi frequency is independent of the ionization rate, which will influence the saturation PI probability only. The larger the ionization rate is, the higher the saturation PI probability will be. These theoretical research results provide a useful guide for the practical application of PI spectroscopy.

Dynamics of the moments of angular momentum distribution during generation of laser-induced polarization spectroscopy (LIPS) signal

The effects of laser-induced anisotropy on the generation of the short-pulse laser-induced polarization spectroscopy (LIPS) signal is investigated by direct numerical integration (DNI) of the time-dependent density matrix equations. The calculations are performed in the short-pulse regime (laser pulse width τL less than characteristic collision time τC) to reduce the influence of collisions on the generation of medium anisotropies. The Zeeman-state structure of the upper and lower energy levels is included in the multistate formulation of the density matrix equations. For a P-branch transition when the isotropic ground-level population is pumped by a circularly polarized light, oriented anisotropy is mainly responsible for the LIPS signal generation; whereas, when the resonance is pumped by a linearly polarized light, aligned anisotropy is mainly responsible. For a Q-branch transition that is pumped by a circularly polarized light, the contributions to the LIPS signal from orientation and alignment are comparable. The effects of saturation on the induced anisotropy is also investigated. The magnitude of the LIPS signal increases by more than a factor of 14 for an initial right circularly polarized-oriented anisotropic distribution in the ground Zeeman states as opposed to an isotropic distribution. An understanding of the effects of anisotropy on the LIPS signal will aid the modeling of the LIPS signal-generation processes and the interpretation of experimental LIPS signals.

Electronic-resonance-enhanced coherent anti-Stokes Raman scattering of nitric oxide: Saturation and Stark effects

A theoretical analysis of electronic-resonance-enhanced (ERE) coherent anti-Stokes Raman scattering (CARS) of NO is described. The time-dependent density-matrix equations for the nonlinear ERE-CARS process are derived and manipulated into a form suitable for direct numerical integration. In the ERE-CARS configuration considered in this paper, the pump and Stokes beams are far from electronic-resonance. The visible 532 and 591 nm laser beams are used to excite Q-branch Raman resonances in the vibrational bands of the X (2)Pi electronic state of NO. An ultraviolet probe beam at 236 nm is used to excite P-, Q-, or R-branch transitions in the (v'=0, v"=1) band of the A (2)Sigma(+)-X (2)Pi electronic system of NO molecule. Experimental spectra are obtained either by scanning the ultraviolet probe beam while keeping the Stokes frequency fixed (probe scans) or by scanning the Stokes frequency while keeping the probe frequency fixed (Stokes scans). The calculated NO ERE-CARS spectra are compared with experimental spectra, and good agreement is observed between theory and experiment in terms of spectral peak locations and relative intensities. The effects of saturation of the two-photon Raman-resonant Q-branch transitions, the saturation of a one-photon electronic-resonant P-, Q-, or R-branch transitions in the A (2)Sigma(+)-X (2)Pi electronic system, and the coupling of these saturation processes are investigated. The coupling of the saturation processes for the probe and Raman transitions is complex and exhibits behavior similar to that observed in the electromagnetic induced transparency process. The probe scan spectra are significantly affected by Stark broadening due to the interaction of the pump and Stokes radiation with single-photon resonances between the upper vibration-rotation probe level in the A (2)Sigma(+) electronic levels and vibration-rotation levels in higher lying electronic levels. The ERE-CARS signal intensity is found to be much less sensitive to variations in the collisional dephasing rates under saturation conditions.

Theory of femtosecond coherent anti-Stokes Raman scattering spectroscopy of gas-phase transitions

A theoretical analysis of coherent anti-Stokes Raman scattering (CARS) spectroscopy of gas-phase resonances using femtosecond lasers is performed. The time-dependent density matrix equations for the femtosecond CARS process are formulated and manipulated into a form suitable for solution by direct numerical integration (DNI). The temporal shapes of the pump, Stokes, and probe laser pulses are specified as an input to the DNI calculations. It is assumed that the laser pulse shapes are 70 fs Gaussians and that the pulses are Fourier-transform limited. A single excited electronic level is defined as an effective intermediate level in the Raman process, and transition strengths are adjusted to match the experimental Raman polarizability. The excitation of the Raman coherence is investigated for different Q-branch rotational transitions in the fundamental 2330 cm(-1) band of diatomic nitrogen, assuming that the pump and Stokes pulses are temporally overlapped. The excitation process is shown to be virtually identical for transitions ranging from Q2 to Q20. The excitation of the Raman coherences is also very efficient; for laser irradiances of 5x10(17) W/m2, corresponding approximately to a 100 microJ, 70 fs pulse focused to 50 microm, approximately 10% of the population of the ground Raman level is pumped to the excited Raman level during the impulsive pump-Stokes excitation, and the magnitude of the induced Raman coherence reaches 40% of its maximum possible value. The theoretical results are compared with the results of experiments where the femtosecond CARS signal is recorded as a function of probe delay with respect to the impulsive pump-Stokes excitation.

Transient coherence oscillation induced by a detuned Raman field in a rubidiumΛsystem

The temporal evolution of coherent population trapping (CPT) was observed in rubidium atomic vapor when sudden changes were made to the detuning of a weak Raman field. The subsequent creation and destruction of CPT are caused by the temporal oscillations of optically induced Raman coherence, with their period depending on the Raman detuning. The oscillating signal was observed over a time of order of tens of milliseconds, and the dependence of the relaxation time on the cell temperature and laser power were investigated. The main features of the experimental observations were well explained by the time-dependent density-matrix equations.

Mid-infrared polarization spectroscopy of carbon dioxide

The use of mid-infrared polarization spectroscopy (PS) for the detection of CO2 has been demonstrated. The P(13) and P(14) resonances of the (0 00 0)→(1 00 1) transition of CO2 were probed using a single-mode optical parametric generator system to produce a high-intensity laser beam at approximately 2.7 μm. The experiments were performed in an atmospheric pressure CO2 jet and also in a sub-atmospheric pressure gas cell. The experimental results were compared with the results of the time-dependent density-matrix equations using direct numerical integration. The Zeeman-state structure of the upper and lower energy levels was included in the multi-state formulation of the density-matrix equations. Fifty-eight Zeeman states and two bath levels were included in the numerical analysis of the P(14) transition. The measured and calculated PS line shapes were in good agreement, and the absolute experimental signal level agreed with the theoretical calculation to within a factor of five.

Polarization spectroscopy using short-pulse lasers: Theoretical analysis

The physics of short-pulse polarization spectroscopy (PS) and the diagnostic potential for quantitative measurements of species concentration are investigated by direct numerical integration (DNI) of the time-dependent density matrix equations for a multistate system. The effects of laser power, collision rates, and Doppler broadening on the short-pulse PS signal generation process are investigated by systematically varying these parameters in the numerical calculations. It is found that the use of a short-pulse laser (laser pulse width τL<characteristic collision time τC) significantly decreases the collision-rate dependence of the PS signal compared to the long-pulse laser case (τL>τC), even for a nonsaturating pump beam. For a saturating pump beam, the short-pulse PS signal is found to be nearly independent of collision rate. Increasing the collision rate by a factor of 100 (from 108 to 1010 s−1), the calculated PS signal strength decreases by only a factor of 2 for a 100-ps pump laser at high intensity. This insensitivity of the PS signal to the collision rate in the medium enhances greatly the potential for quantitative application of the technique for concentration measurements in reacting flows. The underlying physics of the short-pulse PS is explored by studying the effects of collision rate, Doppler broadening, and the pump laser intensity on the temporal profile of the Zeeman state populations and the coherences between the Zeeman states.

Multi-axial-mode laser effects in polarization spectroscopy

The effects of multifrequency-mode laser radiation on polarization-spectroscopy signal generation are investigated by direct numerical integration of the time-dependent density-matrix equations. The numerical solution of the density-matrix equations allows us to incorporate a physically reasonable model for pulsed dye-laser radiation in our analysis of the laser–resonance interaction. The laser radiation is modeled as the sum of electric fields from a finite number of modes that are assumed to have random pulse-to-pulse phases and exponentially distributed amplitudes. Calculations are performed for a homogeneously broadened resonance (only collisional broadening) and for a resonance that is both collision and Doppler broadened. The effect of the multimode laser radiation on polarization-spectroscopy line shapes and saturation curves is investigated for different values of the laser bandwidth and mode spacing and resonance collision and Doppler widths. The saturation parameter for the resonance is strongly dependent on the ratio of the laser bandwidth to the resonance collision width when the laser bandwidth is greater than the collision width. The pulse-to-pulse fluctuations in polarization-spectroscopy signal levels are found to decrease substantially for saturating pump intensities. The inclusion of the multimode laser structure into our density-matrix equations represents a significant advance in modeling the nonlinear interaction of laser radiation with atomic or molecular resonances.

Collisional dependence of polarization spectroscopy with a picosecond laser

The collisional dependence of polarization spectroscopy (PS) with a picosecond-pulse laser is investigated theoretically with a perturbative treatment and experimentally by probing hydroxyl (OH) in a flow cell with a buffer gas of argon. Using a frequency-doubled distributed-feedback dye laser (DFDL), the PS signal strength is monitored as a function of pressure using a nonsaturating pump beam and a saturating pump beam. The collisional dependence of the PS signal is found to decrease significantly with a saturating pump beam. Increasing the flow-cell pressure by a factor of 50 (from 10 torr to 500 torr), the PS signal strength produced with a nonsaturating pump beam decreases by a factor of 18 while that produced with a saturating pump decreases by only a factor of 3. A third-order perturbative (weak-field) approach is used to develop an analytical expression for the PS signal generated by single-mode, exponentially decaying laser pulses. This expression correctly predicts the experimental results acquired with the nonsaturating pump beam. The analytical solution is used to examine the effects of pulse length on the collisional dependence of the weak-field PS signal strength. Results are also presented for a numerical simulation of the time-dependent density matrix equations for the high intensity case.

Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: Saturation and polarization effects

The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.

Theoretical calculation of line shapes and saturation effects in polarization spectroscopy

The physics of polarization spectroscopy (PS) is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The numerical solution of the time-dependent density matrix equations enables us to investigate the effects of strong saturation on PS signal levels and line shapes. Bath levels not directly coupled by the laser radiation are included in the numerical modeling to investigate the effects of collisional rates and different types of collisions on signal levels and line shapes. The effects of Doppler broadening are included by solving the density matrix equations for numerous velocity groups. At low laser power we find that the homogeneously broadened PS line shape is Lorentzian-cubed, as compared to the Lorentzian predicted in several previous low-power analytical solutions. In the low laser power regime, the line-center PS signal is proportional to (collision rate)−6, obviously greatly complicating the application of unsaturated PS for quantitative concentration measurements in flames and plasmas. As the transition begins to saturate at higher laser intensities, the dependences of the signal strength on the laser intensity and on the collision rate decrease drastically, although the line-center PS signal is still approximately proportional to (collision rate)−2. The dependence of the PS signal intensity on the ratio of the population-transfer collision rate to the dephasing collision rate is minimized for saturating pump beam intensities. For resonances that are both Doppler- and collision-broadened, the low-power PS line shape is Lorentzian with a linewidth equal to the collisional width for the case where the Doppler width is much greater than the collisional width. At low pump laser intensities, the PS signal is very dependent on the ratio of Doppler broadening to collisional broadening when the Doppler width is greater than the collisional width. However, at high intensity, the line-center PS signal intensity becomes nearly independent of collision rate when the collisional linewidth is less than the Doppler linewidth. Quantitative application of polarization spectroscopy for concentration measurements in flames and plasmas will almost certainly require resolution of the PS line shape and/or accurate measurement of the saturation curve.

Theoretical investigation of the forward phase-matched geometry for degenerate four-wave mixing spectroscopy

We examine theoretically the degenerate four-wave mixing (DFWM) signal intensities and line shapes obtained with the forward phase-matched geometry in which all beams propagate in the same direction and compare the results to those of the phase-conjugate geometry with counterpropagating pump beams. To examine the forward phase-matched geometry, we modify a theoretical approach used previously to calculate phase-conjugate DFWM signal intensities. This theoretical approach, which involves numerical integration of the time-dependent density-matrix equations, is validated for the forward phase-matched geometry by comparison of our calculated line shapes to both a perturbative solution and to experimental data. This methodology is then used to compare the signal intensities and line shapes obtained with the forward phase-matched geometry and the phase-conjugate geometry in the perturbative (low laser power) and saturated (high laser power) regimes. In the perturbative regime the forward phase-matched signal exhibits less sensitivity to the Doppler linewidth. At pump laser intensities approximately equal to the saturation intensity the signal for the forward phase-matched geometry is stronger than that for the phase-conjugate geometry for primarily Doppler-broadened resonances, assuming the same probe volume for both geometries. These advantages warrant further investigations employing the forward phase-matched configuration for DFWM measurements of gas-phase species.

Interaction of closely spaced resonances in degenerate four-wave-mixing spectroscopy

The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.

Degenerate four-wave mixing spectroscopy with short-pulse lasers: theoretical analysis

The use of picosecond lasers for degenerate four-wave mixing (DFWM) diagnostics of atmospheric pressure flames has been suggested as a means of overcoming the complicated collision-rate dependence of the DFWM signal in the nanosecond-laser pulse-length regime. In this paper DFWM spectroscopy in the regime in which the laser pulse length (τL) is less than the characteristic collisional time (τC) is investigated theoretically. DFWM signal levels for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integration of the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and by summing of the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision broadened and Doppler broadened are considered. DFWM signal generation for these two types of resonances in the fully transient regime (τL≪τC) is strikingly different. For a homogeneously broadened resonance, the DFWM signal persists long after the laser beam has passed through the medium. The decay rate of the DFWM signal is determined by the collision rate in the medium, and the integrated DFWM signal is still dependent on the collision rate even in this fully transient regime. However, when the Doppler broadening of the resonance is much greater than the collisional broadening, the DFWM signal decays with a fall time close to that of the laser pulse. For τL≪τC, we find that it is the laser spectral width, rather than the collisional width, that dictates the range of velocity groups with which the pulses effectively interact. The signal decays with a fall time close to that of the laser pulse because the excited velocity groups radiate out of phase with one another. In a medium in which the Doppler width is much greater than the collisional width, the DFWM reflectivity is independent of collision rate for τL≪τC, but the signal level is dependent on the Doppler width. Finally, we investigate the effect of different laser pulse lengths on reflectivity saturation curves for resonances with different levels of Doppler broadening.

Effect of Doppler broadening on quantitative concentration measurements with degenerate four-wave mixing spectroscopy

The effect of Doppler broadening on degenerate four-wave mixing (DFWM) signal intensities in the regime of high pump and probe laser intensities is investigated theoretically. DFWM reflectivities are calculated by solving the time-dependent density-matrix equations for a two-level system interacting with three laser fields. The density-matrix equations are integrated directly in the time domain on a grid of spatial locations along the phase-matching axis; the DFWM signal level is then calculated by summation of the polarization contribution (with the appropriate phase factor) from each of the spatial grid points. For the case in which the Doppler and the collisional linewidths are comparable, the DFWM reflectivity is found to be inversely proportional to the factor 1 + (bΔωD/ΔωC)2, where ΔωD is the Doppler width, ΔωC is the collisional width, and b is weakly dependent on the pump and the probe laser powers. We developed an analytical expression for the reflectivity of a line that is both collision and Doppler broadened by dividing the widely used Abrams and Lind expression for homogeneous reflectivity Rhom by the factor 1 + (bΔωD/ΔωC)2. This modified reflectivity expression is found to give accurate results for the DFWM reflectivity over a wide range of values for the ratio of Doppler to collisional width. With this modified Abrams–Lind expression, strategies for quantitative DFWM concentration measurements in flames and plasmas are proposed and analyzed. We conclude that, by selection of the appropriate rotational transition, a DFWM reflectivity that is directly proportional to the square of the total species number density can be obtained over a wide range of temperature for constant-laser-intensity spatial profile mapping in flames.

Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations

Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.