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Physical Review A

Relationship between a nonlinear response and relaxation induced by colored noise

D. Leporini

Published Feb 1, 1994• 31 citations
Publisher

Abstract

The nonlinear response to external disturbances of a system of interest scrA (the ``atom'') in contact with a thermal bath is studied. The central issue is the relationship between the nonlinear susceptibility and the equilibrium relaxation spectra of observables of scrA. Paralleling the results of the linear-response theory, the nth-order nonlinear susceptibility is factorized by quantum-mechanical arguments into two terms pertaining to the disturbances and the spin-bath system, respectively. The role of slow atom states, i.e., states relaxing on times T longer than the bath correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$, is pointed out, by recurring to recent findings on their general features [Phys. Rev. A 46, 6222 (1992)]. In the frame of a stochastic picture of the bath and a multilevel scheme of the system scrA, the conditions under which the nonlinear susceptibility provides the spectrum of the slow atom states are clarified. Illustrations are drawn from magnetic resonance to demonstrate that techniques based on the nonlinear response of a spin system to multiple continuous waves compete favorably with pulsed techniques to provide information on longitudinal spin relaxation.

DOI
10.1103/physreva.49.992

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The nonlinear response to external disturbances of a system of interest scrA (the ``atom'') in contact with a thermal bath is studied. The central issue is the relationship between the nonlinear susceptibility and the equilibrium relaxation spectra of observables of scrA. Paralleling the results of the linear-response theory, the nth-order nonlinear susceptibility is factorized by quantum-mechanical arguments into two terms pertaining to the disturbances and the spin-bath system, respectively. The role of slow atom states, i.e., states relaxing on times T longer than the bath correlation time \ensuremathτc{\mathrm{\ensuremath{\tau}}}_{\mathit{c}}\ensuremathτc​, is pointed out, by recurring to recent findings on their general features [Phys. Rev. A 46, 6222 (1992)]. In the frame of a stochastic picture of the bath and a multilevel scheme of the system scrA, the conditions under which the nonlinear susceptibility provides the spectrum of the slow atom states are clarified. Illustrations are drawn from magnetic resonance to demonstrate that techniques based on the nonlinear response of a spin system to multiple continuous waves compete favorably with pulsed techniques to provide information on longitudinal spin relaxation.
In this work, we investigate the influence of colored noise originating from a heat bath on nonlinear Landau-Zener (LZ) tunneling. The nonlinearity might arise intrinsically from many-body mean-field atomic self-interactions or Kerr nonlinearities in optical systems. Contrary to the well-established picture of thermally enhanced quantum tunneling within the linear LZ paradigm, ensemble statistics of our numerical simulations demonstrate that nonlinearity can give rise to thermally suppressed LZ tunneling: the tunneling probability of the nonlinear LZ system decreases as the amplitude of the colored noise increases beyond a critical value. Analytically, the Wiener-Hermite expansion facilitates elucidation of the intrinsic mechanism accounting for the nonmonotonic dependence of tunneling probability on noise amplitude. We compare these behaviors with stochastic resonance responses and clarify that the two phenomena stem from distinct physical mechanisms. In addition, we obtain approximate solutions in the weak- and strong-noise limits and discuss the implications of our theory.
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We investigate the relaxation of multilevel systems under the influence of stochastic fields with finite correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. First, we study a system with spin S=1/2 in a static magnetic field in the presence of dichotomic perturbations. The spectra of the components of the averaged spin vector S\ifmmode\bar\else\textasciimacron\fi{} are evaluated exactly by resorting to the memory-function approach. The model is then extended to include the influence on the relaxation of S\ifmmode\bar\else\textasciimacron\fi{} of a different species with spin I=1/2. The relaxation rates T1{\mathit{T}}_{1}T1​ and T2{\mathit{T}}_{2}T2​ are given in terms of cumulant expansions and compared with the exact results. Fourth-order calculations recover the inequality T2{\mathit{T}}_{2}T2​/2T1{\mathit{T}}_{1}T1​\ensuremath{\le}1 for correlated noise fields. Furthermore, it is found that T1{\mathit{T}}_{1}T1​ can be defined also for long ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. T1{\mathit{T}}_{1}T1​ is longer than ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ provided that ${\mathit{r}}^{(\mathrm{est})}$1, where r(est){\mathit{r}}^{(\mathrm{est})}r(est) is the squared ratio between the total amplitude of the fluctuating fields and the largest one between the Larmor frequency and 1/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. It is also found that 2r(est){\mathit{r}}^{(\mathrm{est})}r(est) is an upper bound of the relative error between the exact and the second-order cumulant expansion of T1{\mathit{T}}_{1}T1​. Generalizations on systems with S=1/2, I>1/2, and arbitrary kinds of noise are discussed.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.