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Physical Review A

Stationary light pulse in solids with long-lived spin coherence

Xiaojun Zhang, Hai-Hua Wang, Lei Wang, Ren-Gang Wan, Jun Kou, Yun-Fei Fan, Bing Zhang, Jin-Yue Gao

Published Jun 7, 2011• 4 citations
Publisher

Abstract

We present a detailed analysis of stationary light pulses (SLP's) for the case of inhomogeneous broadening in both optical and spin transitions, which is normally found in solid materials with long-lived spin coherence. By solving the Langevin equations of motion for the density matrix elements under the integral over the entire range of the inhomogeneous broadenings, the necessary conditions for creating the SLP in a solid are obtained. Then the decay and diffusion processes that the SLP undergoes are analyzed. The characteristics of such processes are studied based on the analytic solution of the SLP with a slowly varying envelope. The dependence of SLP lifetime on inhomogeneous broadenings of spin and optical transitions, which can be regarded as the laser linewidth in the repump scheme, has been discussed.

DOI
10.1103/physreva.83.063804

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We present a detailed analysis of stationary light pulses (SLP's) for the case of inhomogeneous broadening in both optical and spin transitions, which is normally found in solid materials with long-lived spin coherence. By solving the Langevin equations of motion for the density matrix elements under the integral over the entire range of the inhomogeneous broadenings, the necessary conditions for creating the SLP in a solid are obtained. Then the decay and diffusion processes that the SLP undergoes are analyzed. The characteristics of such processes are studied based on the analytic solution of the SLP with a slowly varying envelope. The dependence of SLP lifetime on inhomogeneous broadenings of spin and optical transitions, which can be regarded as the laser linewidth in the repump scheme, has been discussed.
We generalize the theory of electromagnetically induced transparency (EIT) and slow group velocity for the case of the homogeneous and inhomogeneous line broadening in both one- and two-photon transitions which unavoidably takes place in solid materials with a long-lived spin coherence. We identify regimes of EIT where the linewidth can be essentially reduced due to inhomogeneous broadening and, moreover, can be proportional to the amplitude of the driving field rather than the intensity. We suggest also a class of solid materials, namely, rare-earth ion doped semiconductors or dielectrics with electricdipole allowed transitions, that is very promising for realization and applications of EIT.
Analysis on direct transforming the slow light into the stationary light pulse (SLP) is presented. Without the process of turning the slow light into the coherence between the lower levels, the generation of SLP is more efficient. The Maxwell-Liouville equations are employed to study the light pulse dynamics in the samples coherently driven by a bichromatic standing wave. The solution indicates that the forward and backward components of the SLP approach to each other exponentially. Such property is described by introducing a quantity called characteristic length, which appeared to be the ratio of the optical coherence decay rate and the coefficient of the driving term in Maxwell equation. The necessary length for completing the conversion can be estimated as five times the characteristic length. Several materials are analyzed here, including the 87{}^{87}87Rb atoms with different densities, Pr3+{}^{3+}3+-doped yttrium orthosilicate (Pr:YSO), and nitrogen-vacancy color centers in diamond (N-V centers). The values of the characteristic length for the corresponding materials are calculated, as well as the necessary length.
The problem of a light pulse propagating in a nonlinear laser medium is investigated. The electromagnetic field is treated classically and the active medium consists of thermally moving atoms which have two electronic states with independent decay constants ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ in addition to the decay constant ${\ensuremath{\gamma}}_{\mathrm{ab}}$ describing the phase memory. The self-consistency requirement that the field sustained by the polarized medium be equal to the field inducing the polarization leads to coupled equations of motion for the density matrix, and equations of propagation for the electromagnetic field. Although the theory is developed for a Doppler-broadened gaseous medium, it may also be applied to a solid medium with inhomogeneous broadening. A unified treatment is given encompassing a wide range of pulse durations from cw signals to psec pulses. Continuous pumping is allowed, as well as any amount of detuning of the carrier frequency of the pulse from the atomic resonance frequency. The three independent decay constants ${\ensuremath{\gamma}}_{a}$, ${\ensuremath{\gamma}}_{b}$, and ${\ensuremath{\gamma}}_{\mathrm{ab}}$ provide greater flexibility than that obtained by using 1T1\frac{1}{{T}_{1}}T1​1​ and 1T2\frac{1}{{T}_{2}}T2​1​. The equations are solved analytically in a few specialized cases and numerically in the general case. Flow charts for accomplishing the numerical integration are given. Among the special problems considered is the apparent paradox of pulses propagating faster than the velocity of light under circumstances described by Basov et al. It is shown that this contradiction with relativity arises from the use of an unphysical initial condition.
We develop a general scheme for studying the optical response of ultracold atoms driven into a regime of standing-wave electromagnetically induced transparency. We rely on full numerical solutions of the Maxwell-Liouville equations without invoking secular and adiabatic approximations and arbitrary initial state assumptions. These approximations and assumptions can conceal, e.g., significant loss and diffusion responsible for the decay of stationary light pulses in cold atomic samples. The complex decay dynamics of a stationary light pulse is here analyzed in terms of higher-order spin and optical coherences that arise from nonlinear interactions of the stationary light pulse with the two counterpropagating components of a standing-wave driving field. Specific results for stationary light pulses in cold 87Rb^{87}\mathrm{Rb}87Rb atoms have been discussed for temperature regimes where the residual Doppler broadening is negligible.
Recent advances in laser technology have led to the production of ultrashort electromagnetic pulses of very high peak power. In this paper we develop a theory which describes the transient behavior of such pulses in a laser amplifier. The effects of atomic coherence and inhomogeneous broadening, necessary for a complete treatment of the problem, are included. In the present development, the degree of homogeneous and inhomogeneous broadening is variable. The two-level active atoms are characterized by a phase memory time T2{T}_{2}T2​ and an inhomogeneous frequency distribution (which leads to a reversible decay time T2∗T_{2}^{}{}_{}{}^{*}T2​​∗). The field (Maxwell) and atomic (Schr\"odinger) equations are coupled in a self-consistent manner as in Lamb's theory of an optical maser. The resulting equations are solved analytically in certain limits, while the general case is integrated numerically. The emphasis is placed on the situation in which T2∗T_{2}^{}{}_{}{}^{*}T2​​∗ and the pulse width are much shorter than T2{T}_{2}T2​. Thus our main effort is devoted to the physics of the inhomogeneously broadened system. The theory itself, however, contains the effects of both forms of broadening. The basic difference between inhomogeneous and homogeneous broadening is discussed. In the former case, the decay of radiation is caused by the dephasing of dipoles, and is reversible. The atomic memory is retained during the dephasing, and an internal reflection of the atomic coordinates, as is exemplified in the "photon echo" process, will cause the dipoles to rephase. The phonon interruptions or atomic collisions that bring about the homogeneous broadening are random processes which lead to an irreversible destruction of the atomic phase memory. The implications of the reversible versus irreversible decay processes are seen to have important consequences in the problem of pulse amplification.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations that are used to describe the optical pulse propagation and interaction with inhomogeneously broadened two-level medium.The convergence and stability of this numerical method are proved by comparing with the analytic solutions and area theorem derived under special condition.The simulations of arbitrary conditions is discussed by employing this method and analyze the pulse evolution for different detunings,relaxation times and inhomogeneously broadened linewidthes.The established numerical method provides a good way to have further study about the coherent interaction of short optical pulses with resonant mediums,and can be used to solve the corrections of Maxwell-Bloch equations.