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arXiv:1703.01954

Bloch-Siegert Shift and its Kramers-Kronig Pair

Arnab Chakrabarti, Rangeet Bhattacharyya

Published Mar 6, 2017• 0 citations
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Abstract

We report that the Bloch-Siegert shift which appears in Nuclear Magnetic Resonance (NMR) spectroscopy can also be shown to originate as a part of a complex drive-induced second-order susceptibility term. The shift terms thus obtained are shown to have an absorptive Kramers-Kronig pair. The theoretical treatment involves a finite time-propagation of a nuclear spin-$1/2$ system and the spin-bearing molecule under the action of thermal fluctuations acting on the latter. The finite propagator is constructed to account for many instances of thermal fluctuations occurring in a time-scale during which the spin density matrix changes infinitesimally. Following an ensemble average, the resulting quantum master equation directly yields a finite time-nonlocal complex susceptibility term from the external drive, which is extremely small but measurable in solution-state NMR spectroscopy. The dispersive part of this susceptibility term originating from the non-resonant component of the external drive results in the Bloch-Siegert shift. We have verified experimentally the existence of the absorptive Kramers-Kronig pair of the second-order shift term, by using a novel refocussed nutation experiment. Our method provides a single approach to explain both relaxation phenomena as well as Bloch-Siegert effect, which have been treated using non-concurrent techniques in the past.

DOI
10.48550/arxiv.1703.01954

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We report that the Bloch-Siegert shift which appears in Nuclear Magnetic Resonance (NMR) spectroscopy can also be shown to originate as a part of a complex drive-induced second-order susceptibility term. The shift terms thus obtained are shown to have an absorptive Kramers-Kronig pair. The theoretical treatment involves a finite time-propagation of a nuclear spin-1/21/21/2 system and the spin-bearing molecule under the action of thermal fluctuations acting on the latter. The finite propagator is constructed to account for many instances of thermal fluctuations occurring in a time-scale during which the spin density matrix changes infinitesimally. Following an ensemble average, the resulting quantum master equation directly yields a finite time-nonlocal complex susceptibility term from the external drive, which is extremely small but measurable in solution-state NMR spectroscopy. The dispersive part of this susceptibility term originating from the non-resonant component of the external drive results in the Bloch-Siegert shift. We have verified experimentally the existence of the absorptive Kramers-Kronig pair of the second-order shift term, by using a novel refocussed nutation experiment. Our method provides a single approach to explain both relaxation phenomena as well as Bloch-Siegert effect, which have been treated using non-concurrent techniques in the past.
Drive-induced shifts, such as ac Stark shift and Bloch-Siegert shift, are routinely used in various spectroscopies. These shifts are experimentally known to show dispersive Lorentzian behavior as a function of its characteristic frequencies in optical pumping experiments. However, the drive-induced Stark shifts, as calculated using Floquet or dressed atom approaches, do not show the above nonlinear behavior. To address this, we theoretically investigated the drive-induced shifts using a previously reported fluctuation-regulated quantum master equation [A. Chakrabarti and R. Bhattacharyya, Phys. Rev. A 97, 063837 (2018)]. The shifts are obtained as closed-form expressions over the entire detuning range of the drive. The predicted shifts match satisfactorily with the known experimental data of the light shifts. We show that the calculated shifts are a Kramers-Kronig pair of the drive-induced dissipation in conformity with experimental findings. Moreover, we show that at low temperatures, i.e., for less thermal fluctuations, our results asymptotically match with the known theoretical form of the shifts. In the high-temperature regime, we predict that the shifts decrease in magnitude and are inversely proportional to the square of the temperature.
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.
We study the classic problem concerning the dissipative dynamics of a system of two interacting spin-1/21/21/2 particles, coupled to a thermal bath. In particular, we consider the case of a resonant excitation of the initial single-spin coherences and examine the consequences of the interplay between this excitation and the spin-spin dipolar coupling. Using a fluctuation-regulated Quantum Master Equation (frQME) [Phys. Rev. A 97, 063837 (2018)] we show that the relevant dissipator consists of cross-terms between the drive and the dipolar Hamiltonian, apart from regular self-terms responsible for ordinary relaxation effects. The drive-dipole cross-correlations provide for a novel second-order coupling of single and two-spin observables in the Zeeman basis, which cannot be classified as a dissipative effect. We show that the presence of these unique second-order non-dissipative terms lead to the well-known spin-locked steady-states in dipolar spin-ensembles. Importantly, the mathematical form of the steady-state magnetization in the spin-locked phase, obtained from our theory, matches with the prediction of the traditional spin-temperature based methods used to describe such phenomena.
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.