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 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.
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 dynamics of multiphoton transitions in a two-level spin system excited by transverse microwave and longitudinal RF fields with the frequencies ω mw and ω rf , respectively, is analyzed. The effective time-independent Hamiltonian describing the “dressed” spin states of the “spin + bichromatic field” system is obtained by using the Krylov-Bogoliubov-Mitropolsky averaging method. The direct detection of the time behavior of the spin system by the method of nonstationary nutations makes it possible to identify the multiphoton transitions for resonances ω 0 = ω mw + r ω rf (ω 0 is the central frequency of the EPR line, r = 1, 2), to measure the amplitudes of the effective fields of these transitions, and to determine the features generated by the inhomogeneous broadening of the EPR line. It is shown that the Bloch-Siegert shifts for multiphoton resonances at the inhomogeneous broadening of spectral lines reduce only the nutation amplitude but do not change their frequencies.
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.
For pt.I see Chem. Phys. Lett. vol.96, p.464 (1983). The non-linear multiphoton excitation dynamics of two-level systems under the influence of two strong linearly polarised monochromatic fields is studied exactly for the first time along the semiclassical many-mode Floquet theory. The present approach relaxes the restrictions of the conventional generalised rotating-wave approximation (GRWA) and allows correct treatment of various non-linear features such as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending the nearly degenerate perturbative method of Salwen (1955), approximate analytical expressions for transition probabilities are obtained for multiphoton resonance processes for which (n+1) photons of one field are absorbed, whereas n photons of the other field are emitted, where n is an arbitrary integer. Detailed comparison of the analytical, the GRWA and the numerically exact results is given. Also presented is a case study of the time evolution of spin 1/2 systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
We have studied multiple-photon transition in a low-field nuclear magnetic resonance (NMR) experiment. It is well known that a saturation of the NMR occurs when using a sufficiently high alternating magnetic field which induces multiple-photons transitions. We show theoretically that these multiphoton transitions are accompanied by an harmonic generation. For an-photon transition, this generation occurs mostly on thenth and (n \pm 1)th harmonic. For an irradiating field2H_{1} \cos \omegat of pulsation ω such thatn\omega= \gamma H_{0}=\omega_{0}where H0is the steady field and θ is the angle between H0and H1the amplitudesS\min{n}\max{n}andS\min{n}\max{n+1}of these components are given byS\min{n}\max{n} = f_{n}(\theta) (H_{1})^{n},S\min{n}\max{n+1} = \cos \theta f_{n}(\theta) (H_{1})^{n+1}. We have verified these equations in a low-field NMR experiment (H_{0} = 0.7Gs/s), polarizing first a flowing liquid in a high magnetic field. The liquid then flows in a modified Bloch spectrometer. The receiving coil is still perpendicular to the steady field but it is possible to adjust the angle θ between the axis of the emission coils (H1field) and the steady-field H0. Using a synchronous detection at the output of the receiving coil on the pulsation(n - 1) \omega, n\omega, or(n + 1) \omega, we have directly detected multiphoton transistions. The above equations have been verified forn = 2ton = 5. There are slight discrepancies at high excitation which may be explained taking into account a large saturation.