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Physical Review A

Selective spin inversion in nuclear magnetic resonance and coherent optics through an exact solution of the Bloch-Riccati equation

Michael S. Silver, R. I. Joseph, David I. Hoult

Published Apr 1, 1985• 401 citations
Publisher

Abstract

An analytical solution of the Bloch equations is presented using as a driving function a complex hyperbolic secant pulse. The solution indicates that, under the appropriate conditions, the use of such a pulse creates a highly selective population inversion which, above a critical threshold, is independent of pulse (irradiating ${B}_{1}$ field) amplitude and hence ${B}_{1}$ field homogeneity. This result is shown to be in excellent agreement with experiment, and as an aid to comprehension, analogies to adiabatic rapid passage and self-induced transparency are drawn.

DOI
10.1103/physreva.31.2753

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The Journal of Chemical Physics•Aug 15, 1989•25 citations
An analytical solution of the Bloch equations is presented using as a driving function a complex hyperbolic secant pulse. The solution indicates that, under the appropriate conditions, the use of such a pulse creates a highly selective population inversion which, above a critical threshold, is independent of pulse (irradiating B1{B}_{1}B1​ field) amplitude and hence B1{B}_{1}B1​ field homogeneity. This result is shown to be in excellent agreement with experiment, and as an aid to comprehension, analogies to adiabatic rapid passage and self-induced transparency are drawn.
An analytic solution of the Bloch equation is presented using a class of asymmetric complex functions as a driving function. These may involve both amplitude and frequency modulation functions. Analytic solutions that have been previously derived include the amplitude modulated (``real'') hyperbolic secant pulse and its asymmetric generalization in addition to the complex sech/tanh pulse. The solution presented in this paper contains all of these as particular cases. The solution in its complex form is closely associated with an adiabatic fast passage experiment and renders robust magnetization inversion provided the rf amplitude exceeds a certain threshold. \textcopyright{} 1996 The American Physical Society.
In this research study, we derive the exact solutions of the Bloch equations describing the dynamics of a two-level atom with dephasing. In the two-level atom, a strong laser pump couples a ground state to an upper excited state with a time-dependent Rabi-frequency. The exact solutions are given for the atomic population inversion and the real and imaginary parts of the coherence while the input pulse is an asymmetric hyperbolic cosine form. Additionally, the system is under a chirped detuning. The method of solving the Bloch equations analytically is a very tedious part of the research, and as far as we know, there are few exact solutions available in this field. Hence, our solutions might be of great interest to various research areas, including nuclear magnetic resonance, where analytical solutions to the Bloch equations play a major role in the study of the information on the state of the medium as determined by the NMR signals.
An exact analytic solution to the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant pulsed external field. The pulse has a hyperbolic-secant shape, with a sign jump (i.e. a phase step of π) at its maximum, which nullifies the overall pulse area. It is shown that off resonance, for detunings larger than the pulse bandwidth, the transition probability approaches unity as the field intensity increases. The population inversion is robust against small-to-moderate variations in the detuning and the intensity, a feature reminiscent of adiabatic passage. The population inversion, however, is not of adiabatic nature; it is induced by δ-function-shaped nonadiabatic coupling with an area of nearly π. An estimate of the required experimental resources shows that implementation with shaped femtosecond pulses is feasible.
Doppler-free population inversion induced by coherent two-photon resonance is calculated analytically for a two-photon analog of the hyperbolic-secant pulse and numerically for a Gaussian, linearly chirped pulse. The numerical calculations are carried out for a two-photon resonance of a chirped ruby laser line with the $6{S}_{\frac{1}{2}}\ensuremath{\rightarrow}9{D}_{\frac{3}{2}}$ transition in Cs. Complete inversion is found when the fractional change in frequency during the pulse lies between 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$. The coherent two-photon resonances are described by optical Bloch equations which are derived from a multiple-time-scale perturbation theory.
The generation of an optical breather in coherent, resonant pulse propagation is studied. The initial field envelope is assumed to be in the form of a hyperbolic secant, with one half inverted so that the total area of the pulse is equal to zero. The Maxwell–Bloch equations for an inhomogeneously broadened atomic line are solved by the inverse-scattering method for all values of the independent variables x and t. The scattering amplitudes are found for any value θ of the absolute area of the initial pulse. For θ = 2πn the reflection coefficients are rational functions of the spectral parameter, and the inverse problem can be solved exactly. It is shown that the x-dependent poles of the reflection coefficients are responsible for the transient pulse shape. Such poles appear only in the presence of inhomogeneous broadening. The threshold for transparency is found at θ = 2π. At this value of the area, a breatherlike decaying solution is obtained. At θ = 4π a breather appears. Its energy amounts to approximately 70% of the initial pulse energy. For θ = 2π and 0 = 4π the solutions are analyzed in detail.
By implementing the Wei–Norman Lie algebra approach, this paper focuses on the development of a new method of solution to the quantum Liouville equation for NMR field modulation. First, the general method is reviewed and discussed for systems of any finite dimensional Lie algebra. The theory is then applied to arbitrary time-dependent Zeeman interactions [or SU(2) Hamiltonians] in which the complete arbitrary spin I≥1/2 density operator problem reduces to a Riccati equation. Novel exact analytical solutions of the complete spin density operator for a class of temporal field modulations are obtained. In particular, a spherical tensor operator basis is used to expand the density operator, and the solutions retain the physically appealing form of Wigner rotations with time-modulated rotation angles which are special functions with well known analytical properties. The exact solutions include the frequency swept hyperbolic secant pulse shapes, as well as any exponentially modulated amplitude pulse. In contrast to all other existing formulations, the present treatment provides the first known examples of exactly soluble nonrectangular pulse shapes for all resonance off-set, all field amplitudes, all time and valid for all spin I≥1/2. In addition, from the underlying Riccati equation, a new series solution for the complete spin density matrix is given which is valid for any well-behaved field modulations. Spin inversion profiles are also calculated for exponentially decaying pulses. These show applications to selective excitation.