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The Journal of Chemical Physics

Observation of three-photon resonances in NMR under four-field irradiation

Gadi Goelman, David B. Zax, Shimon Vega

Published Jul 1, 1987• 22 citations
Publisher

Abstract

Experimentally achievable conditions where only weak off-resonant fields are used to excite coherence in a two-level system are explored. A brief review of previous theoretical results useful in treating multifrequency irradiation sequences is provided. Using an approximation method based on the Floquet formalism we derive an expression for the effective field strength as a function of the phases and amplitudes of the applied irradiation fields. Some of these results are rederived and/or explained in a simplified manner. Under conditions of four-frequency irradiation, three-photon excitation of a two-level system in NMR is experimentally demonstrated on the spin-1/2 system of the 31P nucleus in phosphoric acid. The behavior of the three-photon resonance is explored as a function of all important experimental parameters. The results of these tests conform closely to the theoretical predictions. Possible uses of multifrequency irradiation schemes in NMR are suggested.

DOI
10.1063/1.453630

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Physica Scripta•Jan 1, 1988•1 citation
Experimentally achievable conditions where only weak off-resonant fields are used to excite coherence in a two-level system are explored. A brief review of previous theoretical results useful in treating multifrequency irradiation sequences is provided. Using an approximation method based on the Floquet formalism we derive an expression for the effective field strength as a function of the phases and amplitudes of the applied irradiation fields. Some of these results are rederived and/or explained in a simplified manner. Under conditions of four-frequency irradiation, three-photon excitation of a two-level system in NMR is experimentally demonstrated on the spin-1/2 system of the 31P nucleus in phosphoric acid. The behavior of the three-photon resonance is explored as a function of all important experimental parameters. The results of these tests conform closely to the theoretical predictions. Possible uses of multifrequency irradiation schemes in NMR are suggested.
The use of four pulsed rf fields for the excitation of multiphoton resonances in a two-level system is investigated. A general theoretical description of the experiment is provided based on the Floquet formalism and application of basic conservation laws to the overall transition process. The predictions of the theory are compared with computer simulations of the time evolution of the magnetization according to the exact time-dependent Hamiltonian. It is shown that N-photon resonances occur when the fields are applied at frequencies \ifmmode\pm\else\textpm\fi{}k\ensuremath{\omega} and \ifmmode\pm\else\textpm\fi{}l\ensuremath{\omega} from the Larmor frequency, where k and l are positive integers having no common factors and where 0\ensuremath{\le}k<l, k+l=N. For any resonance k photons are taken from the outer field pair and l are taken from the inner field pair. A total of 2[(l+k)!]/(k!l!) mutually interfering transition pathways exist for the overall process and the resultant magnitude of the effective N-photon field is very sensitive to the relative initial phases of the rf fields. The spiral motion observed for multiphoton resonances induced by double-frequency irradiation is absent here when the field intensities are symmetrical. In addition, the fraction of total applied rf intensity available for multiphoton pumping is considerably greater than for two fields. Phase cycling to enhance detection of multiphoton effects and multiphoton spin locking are considered.
The expression for the absorption coefficient in two-photon resonance transitions is deduced under general conditions. No particular restrictions are imposed on the level system or on the frequencies and intensities of the fields. The following cases are considered: (i) a two-level system (with and without singularities) irradiated by a single wave and (ii) a three-level system irradiated by two waves. Some theoretical and experimental disagreements reported by authors of previous papers are examined and clarified by the present theory.
A general theory has been developed to calculate the response, in NMR conditions, of a spin system irradiated by two nearly resonant waves having different frequencies and intense enough to produce an appreciable saturation. The absorption signal has been calculated employing the statistical operator technique, treating the electromagnetic field by the second quantization formalism. Two kinds of transitions are expected: The first one arises from processes in which n photons of one wave are emitted and n photons of the other wave are absorbed, the second one arises from processes involving the emission of n photons of one wave and the absorption of n ± 1 photons of the other. In the first case all the transitions coalesce in a single dispersionlike signal, which occurs when the frequencies of the two waves are nearly the same; in the second case a many-line spectrum occurs in which emission and absorption lines are present. The experimental results here reported strictly agree with theoretical previsions.
This thesis studies the interaction between two-and three-level systems and strong, coherent radio-frequency (rf) electrom agnetic fields.The investigations were carried out in both the frequency domain, where steady state properties are examined, and the time domain, where transient behaviours are studied.The two-and three-level systems studied arise from the nuclear spin I = 1 triplet within the 3 A ground state of the nitrogen-vacancy (N-V) centre in diamond.The nuclear magnetic resonance (NMR) is detected optically using a coherent optical-rf double resonance Raman heterodyne technique.Owing to the unique features of both the nuclear spin triplet and the detection technique, many interesting experimental results have been obtained.The thesis is comprised of four parts.Part 1 includes Chapters 1 to 3, which provides introductory accounts.The two-level system concept and associated theoretical treatments in the density matrix formalism are introduced in Chapter 1.The theory is given in a very general form and is developed later in relevant chapters for specific situations.Chapter 2 gives an introduction to the N-V centre and its electronic and hyperfine energy levels.The Raman heterodyne technique and the experimental details are described in Chapter 3.In Part 2, consisting of Chapters 4 and 5, the nuclear magnetic transition of interest is characterized.Chapter 4 describes various coherent transient observations including nutation, free-induction-decay (FID), spin echo, spin locking, delayed FID, and rotary echo, from which important parameters such as the transition strength, population decay time T 1? and dephasing time T2 can be deduced.The dependence of these parameters upon the wave function mixing due to the level anti-crossing is discussed in Chapter 5.Part 3 is the most significant part of the thesis and consists of Chapters 6 to 9. In these chapters the interaction of a two-and three-level system with electromagnetic fields are studied in the frequency domain using pump-probe spectroscopy.This technique involves driving the systems with a strong (pump) field and monitoring it with a separate (probe) field.The absorption and dispersion profiles of a driven two-level system probed by a weak field are discussed in Chapter 6. Topics include the dynamic "Stark" effect and amplification without inversion.When the probe intensity is increased from weak to strong, multiphoton transitions and multiphoton gain without inversion are observed.These effects are described in Chapter 7. When the pump and probe fields interact separately with two transitions which share one common level the two-level system is extended to a three-level one.The power broadening of the Autler-Townes spectrum in such a three-level system is the topic presented in Chapter 8. Chapter 9 considers the case where there are two strong fields and a weak probe field.The discussion is in terms of the dressed state picture and it is shown that the dressed state can be also dressed.Experimental results of the transient nutation in dressed states are presented in part 4 which consists of Chapter 10.The dressed states transition strength is determined for the first time experimentally.Finally, Chapter 11 presents a summary of this thesis.
This investigation describes the interaction of a three level atom with two laser fields. One of the transitions from the ground state is in resonance with twice the frequency of the first laser and the other transition from the ground state is in resonance with the second laser. The Green's function formalism is used to derive expressions from which the induced two photon and one photon excitation spectra are computed. Also, approximate expressions are derived for the excitation spectra in the appropriate frequency regions. These results agree well with the numerical computations based upon the precise expressions. The interference between the two transitions produce some splittings; these splittings depend upon the Rabi frequency of the one photon transition. The intensities of the weak peaks depend upon the ratio of the Rabi frequency of the two photon transition to the frequency of the first laser. Some features of the excitation spectra are interpreted in terms of previous knowledge about the behavior of two level atoms in strong laser fields.
Multiphoton coherent interaction effects are investigated in atoms and molecules. It is demonstrated theoretically as well as experimentally that, in the presence of a two photon resonance, maximum conversion efficiency can be achieved using energy preserving pulse sequences, in condition of coherent interaction. However, highest conversion efficiencies are obtained with weak two photon resonances. A study of four photon resonant coherent interaction in mercury points to a peculiar quenching of the interaction by the generated third harmonic. Four photon resonant coherent interaction is also investigated in molecules, where it is shown that an input signal can be found that results in selective excitation of the upper levels.