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arXiv:quant-ph/0001095 · Physical Review Letters

Stochastic Resonance and Nonlinear Response using NMR Spectroscopy

Lorenza Viola, Evan M. Fortunato, Seth Lloyd, Ching-Hsiang Tseng, David G. Cory

Published Jun 12, 2000• 37 citations
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Abstract

We revisit the phenomenon of quantum stochastic resonance in the regime of validity of the Bloch equations. We find that a stochastic resonance behavior in the steady-state response of the system is present whenever the noise-induced relaxation dynamics can be characterized via a single relaxation time scale. The picture is validated by a simple nuclear magnetic resonance experiment on water.

DOI
10.1103/physrevlett.84.5466

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We revisit the phenomenon of quantum stochastic resonance in the regime of validity of the Bloch equations. We find that a stochastic resonance behavior in the steady-state response of the system is present whenever the noise-induced relaxation dynamics can be characterized via a single relaxation time scale. The picture is validated by a simple nuclear magnetic resonance experiment on water.
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The extension of double resonance methods in nuclear magnetic resonance is presented. The origin of the problem and its reduction to an abstract problem are discussed. The theorem that allows the calculation of the ensemble average of the linear response of a system with a strong time-dependent stochastic perturbation is given and applied to one-spin and two-spin systems with scalar coupling. The question of convergence is discussed and a relation between level shift and line broadening is stated. The nuclear magnetic resonance spectra are described in relation to the shape of the power spectrum of the stochastic perturbation. The example shows that the functional expansion of the density operator is useful to calculate the effects of strong stochastic perturbations on a quantum mechanical system. (M.C.G.)
Nuclear Magnetic Resonance (NMR) experiments involve coherent and incoherent dynamics. Incoherent dynamics give rise to nuclear spin relaxation typically accounted for by a differential equation known as a quantum master equation. Out of the existing master equations the Lindblad equation is particularly important. A central theorem of open quantum theory states that a relaxation process represents a quantum Markovian process if and only if it can be described by a Lindblad master equation. This in turn raises serious questions regarding the compatibility of conventional NMR relaxation theory and one of the most fundamental results of open quantum theory. The NMR treatment of important Markovian relaxation mechanisms such as Brownian motion or chemical exchange events do in general not result in a master equation in Lindblad form and would therefore be at odds with the mathematically rigorous theory of open quantum systems. We explore this apparent conflict for two important chemical exchange phenomena and show that their conventional treatment may be recast into a suitable Lindblad equation. Similarities between spatial diffusion and multi-site exchange events allow us to extend our results to nuclear spin relaxation due to translational diffusion. As a result we are able to show that the NMR treatment of intramolecular exchange and translational diffusion represent valid quantum Markovian processes compatible with the rigorous definitions of open quantum theory.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
Radio wave amplification by stimulated emission of radiation (NMR laser) was observed in Al2{\mathrm{Al}}_{2}Al2​O3{\mathrm{O}}_{3}O3​: Cr3+{\mathrm{Cr}}^{3+}Cr3+ (ruby) with the 27Al^{27}\mathrm{Al}27Al spin system pumped to an inverted Zeeman state by means of dynamic nuclear polarization (DNP) and with an LC circuit tuned to the NMR transitions. A theory is developed which combines standard Bloch equations with the dynamic equations of a heat-reservoir model for the spin dynamics. Experimental data concerning the relaxation and polarization mechanisms in the nonradiant state are presented, and the parameters of the reservoir model are determined. In accord with these results, dynamic NMR laser equations for different modes of operation (continuous or pulsed) are derived from which the threshold of the NMR laser, its critical behavior near threshold, and its transient response after QQQ-switch tuning can be calculated. The agreement between theory and experiments in the superradiant state reflects unambiguously the interplay between nuclear-spin relaxation, DNP, spin fluctuations, and the self-induced ordering of the 27Al^{27}\mathrm{Al}27Al spins perpendicular to the applied field. In particular, the results highlight the decisive role of the electronic dipole-dipole system in the spin and NMR laser dynamics of ruby. Special emphasis is placed on the close relationship between the NMR laser, as a system far from thermal equilibrium which shows the phenomenon of cooperative self-organization, and a system at thermal equilibrium which undergoes a second-order phase transition. Thus, the rotating nuclear-spin magnetization is treated as an order parameter.