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Physical Review A

Coherent and phase-sensitive phenomena of ultrashort laser pulses propagating in three-level Λ -type systems studied with the finite-difference time-domain method

Yurii V. Loiko, Carles Serrat

Published Jun 8, 2006• 23 citations
Publisher

Abstract

Propagation of single- and two-color hyperbolic secant femtosecond laser pulses in a three-level $\ensuremath{\Lambda}$-type quantum system is investigated by solving the Maxwell and density matrix equations with the finite-difference time-domain and Runge-Kutta methods. As a first study of our modeling, we simulate pulse self-induced transparency (SIT) in two-level systems and see how this phenomenon can be controlled by manipulating the initial relative phase between the SIT pulse and a second control pulse, provided the ratio between both pulse frequencies obeys the relation ${\ensuremath{\omega}}_{1}∕{\ensuremath{\omega}}_{2}=3$. We then examine frequency down-conversion processes that are observed with single- and two-color pulses the envelope area of which is equal to or a multiple of $2\ensuremath{\pi}$, for pulse frequencies close to resonance with the transitions of a three-level $\ensuremath{\Lambda}$ medium. Also, phase-sensitive phenomena are discussed in the case of two-color $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ pulses propagating resonantly in the three-level system. In particular, possibilities for such coherent control are found for frequency down-conversion processes when the ratio of the frequencies of optical transitions is ${\ensuremath{\omega}}_{13}∕{\ensuremath{\omega}}_{12}=3$. The conditions for quantum control of four-wave mixing processes are also examined when the pulse frequencies of two-color $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ pulses are far from any resonance of the three-level system. We demonstrate the possibility to cancel the phase sensitivity of the four-wave coupling in a $\ensuremath{\Lambda}$-type system by competition effects between optical transitions.

DOI
10.1103/physreva.73.063809

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Propagation of single- and two-color hyperbolic secant femtosecond laser pulses in a three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type quantum system is investigated by solving the Maxwell and density matrix equations with the finite-difference time-domain and Runge-Kutta methods. As a first study of our modeling, we simulate pulse self-induced transparency (SIT) in two-level systems and see how this phenomenon can be controlled by manipulating the initial relative phase between the SIT pulse and a second control pulse, provided the ratio between both pulse frequencies obeys the relation \ensuremathω1∕\ensuremathω2=3{\ensuremath{\omega}}_{1}∕{\ensuremath{\omega}}_{2}=3\ensuremathω1​∕\ensuremathω2​=3. We then examine frequency down-conversion processes that are observed with single- and two-color pulses the envelope area of which is equal to or a multiple of 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ, for pulse frequencies close to resonance with the transitions of a three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ medium. Also, phase-sensitive phenomena are discussed in the case of two-color \ensuremathω\ensuremath-3\ensuremathω\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}\ensuremathω\ensuremath-3\ensuremathω pulses propagating resonantly in the three-level system. In particular, possibilities for such coherent control are found for frequency down-conversion processes when the ratio of the frequencies of optical transitions is \ensuremathω13∕\ensuremathω12=3{\ensuremath{\omega}}_{13}∕{\ensuremath{\omega}}_{12}=3\ensuremathω13​∕\ensuremathω12​=3. The conditions for quantum control of four-wave mixing processes are also examined when the pulse frequencies of two-color \ensuremathω\ensuremath-3\ensuremathω\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}\ensuremathω\ensuremath-3\ensuremathω pulses are far from any resonance of the three-level system. We demonstrate the possibility to cancel the phase sensitivity of the four-wave coupling in a \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type system by competition effects between optical transitions.
Quantum coherence and interference in atomic systems can lead to interesting optical phenomena such as nonabsorbing resonances, lasing without inversion, generating an ultrahigh index of refraction without absorption, propagation of light pulses without losses. In the last-named is given the title transparency. Up to now it is known some types of transparency in multi-level resonant media. One of them is electromagnetically induced transparency (EIT). The phenomenon of EIT in the propagation of laser radiation through an absorbing medium has been recently investigated and explained in terms of quantum coherence and interference for multi-level atomic system. For instance in the (Lambda) -system, if the atomic medium is prepared in a quantum superposition of states, matched pulses, i.e., a pair of optical pulses whose amplitude and phase have a well defined relation, propagate without absorption. The mechanism of coherent population trapping is responsible for EIT in a (Lambda) -system. There is another, well known mechanism for creating transparency in a two-state atom: self-induced transparency, or SIT. Here, a single pulse whose area (the integral of its Rabi frequency over time) is 2(pi) is applied to the medium. Such a pulse causes the atoms to cycle smoothly from the ground to the excited state and then back again. In principle, such a pulse can propagate over long distances. SIT can be generalized to multi-level systems and multiple quasi-monochromatic fields. Generalization of SIT in a (Lambda) -system was obtained in reference 2. In this report we present a new type of transparency (combined transparency). This type describes propagation of laser pulse through a (Lambda) -type medium where the initial atomic state is prepared in the form of spatially varying coherent superposition of low-energetic states (initial coherency gradient). The common envelope behaves as usual 2(pi) hyperbolic secant pulse, but the ratio between the components of frequencies transitions 1> yields 3> and 2> yields 3> changes due to coherent Raman scattering so (phi) equals atan(E1/E2) equals (pi) /2-Gz, where z is the coordinate, G is the constant determined by both the incident pulse parameters and the initial coherence between the atomic states. Our solution demonstrates a direct connection of EIT in adiabatic regime to the SIT effects. This type of a transparency is given the title screw-type transparency, because for the given form of incident pulse (pitch of screw) medium is transparent, if medium's the initial coherence gradient is correspondingly prepared (thread on a engine nut). Besides a transparency, this phenomenon contains a transformation of the incident pulse frequency during the process of the propagation. This property of the phenomena can be result in a complete conversion of the frequency.
Coherent control of regions with spatial excitation of populations and coherence between two lower states in three-level $\ensuremath{\Lambda}$-type quantum systems mediated by the self-induced transparency (SIT) phenomenon is theoretically investigated with one- and two-color ultrashort pulses and with pulse sequences beyond the slowly varying envelope (SVEA) and rotating wave (RWA) approximations. The effects of different parameters characterizing the pulses and the medium on the location as well as the width of such excited regions have been studied numerically by means of the finite-difference time-domain (FDTD) method. It has been determined that with a scheme of two-pulse excitation one can effectively control the position at which the region is written and its width. In particular, the position of the excited region can be controlled by the area of the pulses. We find that the maximum value of the population transferred to the lower excited state depends on the detuning of the pulses with respect to the one-photon resonances, and that both position and width of the region also depend on the temporal duration of the pulses. We show how after the excited region is written, its position can be shifted by additional pulses. On the basis of numerical results, scaling laws are formulated for the reported phenomena. With such control, the width of the region excited inside the medium can be reduced to the order of the wavelength of the light.
A theoretical investigation on the coherent control of optical transient four-wave mixing interactions in two-level systems with two intense few-cycle propagating laser pulses of central angular frequencies $\ensuremath{\omega}$ and $3\ensuremath{\omega}$ is reported. By numerically solving the full Maxwell-Bloch equations beyond the slowly varying envelope and rotating-wave approximations in the time domain, the nonlinear coupling to the optical field at frequency $5\ensuremath{\omega}$ is found to depend critically on the initial relative phase $\ensuremath{\phi}$ of the propagating pulses: the coupling is enhanced when the pulses interfere constructively in the center $(\ensuremath{\phi}=0)$, while it is nearly suppressed when they are out of phase $(\ensuremath{\phi}=\ensuremath{\pi})$.
The interference resonant propagation of two-color 4π-4π femtosecond Gaussian pulses in three-level Λ-type atomic medium with different values of the relative phase φ is investigated by using numerical solution, which is obtained by the finite-difference time domain (FDTD) method and the iterative predictor–corrector (PC) method for the full Maxwell-Bloch equations. It is found that, for a smaller value of the φ, the pulse splitting occurs, and when the value of the φ increases to a certain value, only the variation of pulse shape present but the pulse splitting no longer occurs; accordingly, their spectral strengths and frequency ranges of the two-color pulses decrease remarkably with the value of φ increasing; Moreover, the relative phase also has an obvious effect on population, different population evolutions can be achieved by adjusting the value of φ.
A critical dependence of the quantum interference on the optical Stark spectral shift produced when two-color laser pulses interact with a two-level medium is observed. The four-wave mixing of two ultrashort phase-locked $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ laser pulses propagating coherently in a two-level system depends on the pulses' relative phase. The phase dominating the efficiency of the coupling to the anti-Stokes Raman component is found to be determined by the sign of the total ac Stark shift induced in the system, in such a way that the phase sensitivity disappears precisely where the ac Stark effect due to both pulses is compensated. A coherent control scheme based on this phenomenon can be contemplated as the basis for nonlinear optical spectroscopy techniques.
We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell's curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. We introduce a dipole-coupling interaction Hamiltonian allowing Rabi flopping of the population difference along and perpendicular to the propagation axis with frequencies depending on the corresponding field components. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system is derived in order to couple the quantum system equations to the Maxwell's curl equations. The pseudospin equations are phenomenologically extended to include relaxation effects by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.