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Physical Review A

Transparency in two-level spin systems induced by a longitudinal field

Moritz Kälin, Igor Gromov, Arthur Schweiger

Published Mar 17, 2004• 19 citations
Publisher

Abstract

In an electron-spin two-level system a bichromatic radiation field, consisting of a transverse microwave field and a radio frequency field oriented parallel to the static magnetic field, can induce multiple photon transitions of the type ${\ensuremath{\sigma}}_{\text{mw}}^{+}+k{\ensuremath{\pi}}_{\text{rf}}$, with the photons ${\ensuremath{\sigma}}_{\text{mw}}^{+}$ and ${\ensuremath{\pi}}_{\text{rf}}$ absorbed from the microwave and radio frequency fields, and $k$ being an integer. The two-level spin system may become transparent under such a bichromatic radiation field. This phenomenon directly depends on the presence of the $\ensuremath{\pi}$ photons of the longitudinal radio frequency field. An analytical description of the multiple photon processes and the resulting transparency effects, based on a toggling-frame approach, is given. The effective field amplitudes of the multiple photon transitions are found to have a Bessel-function-like dependence on the radio frequency amplitude. The behavior of the magnetization vector during a bichromatic pulse is illustrated by using classical equations of motion. The theoretically predicted effects are verified experimentally. As an example for an application of the investigated type of electromagnetically induced transparency, a one-pulse echo experiment is described, where the free evolution period is created artificially.

DOI
10.1103/physreva.69.033809

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In an electron-spin two-level system a bichromatic radiation field, consisting of a transverse microwave field and a radio frequency field oriented parallel to the static magnetic field, can induce multiple photon transitions of the type \ensuremathσmw++k\ensuremathπrf{\ensuremath{\sigma}}_{\text{mw}}^{+}+k{\ensuremath{\pi}}_{\text{rf}}\ensuremathσmw+​+k\ensuremathπrf​, with the photons \ensuremathσmw+{\ensuremath{\sigma}}_{\text{mw}}^{+}\ensuremathσmw+​ and \ensuremathπrf{\ensuremath{\pi}}_{\text{rf}}\ensuremathπrf​ absorbed from the microwave and radio frequency fields, and kkk being an integer. The two-level spin system may become transparent under such a bichromatic radiation field. This phenomenon directly depends on the presence of the \ensuremathπ\ensuremath{\pi}\ensuremathπ photons of the longitudinal radio frequency field. An analytical description of the multiple photon processes and the resulting transparency effects, based on a toggling-frame approach, is given. The effective field amplitudes of the multiple photon transitions are found to have a Bessel-function-like dependence on the radio frequency amplitude. The behavior of the magnetization vector during a bichromatic pulse is illustrated by using classical equations of motion. The theoretically predicted effects are verified experimentally. As an example for an application of the investigated type of electromagnetically induced transparency, a one-pulse echo experiment is described, where the free evolution period is created artificially.
EPR transient nutation spectroscopy is used to measure the effective field (Rabi frequency) for multiphoton transitions in a two-level spin system bichromatically driven by a transverse microwave (MW) field and a longitudinal radio-frequency (RF) field. The behavior of the effective field amplitude is examined in the case of a relatively strong MW field, when the derivation of the effective Hamiltonian cannot be reduced to first-order perturbation theory in ω1/ωrf(ω1 is the microwave Rabi frequency, ωrf is the RF frequency). Experimental results are consistently interpreted by taking into account the contributions of second and third order in ω1/ωrf evaluated by Krylov-Bogolyubov-Mitropolsky averaging. In the case of inhomogeneously broadened EPR line, the third-order correction modifies the nutation frequency, while the second-order correction gives rise to a change in the nutation amplitude due to a Bloch-Siegert shift.
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.
Atomic coherence and interference play an important role in the study of the atom-photon interactions. Electromagnetically induced transparency (EIT) is an extensively studied two-photon coherence phenomenon theoretically as well as experimentally. EIT is mainly observed in three-level atomic systems which causes transparency by quenching absorption of the medium. In this paper, based on the lambda type three-level system including energy level |1>, |2> and |3>, a microwave driving field is introduced between the excited-state energy level |3> and another excited-state energy level |4> to form an inverted Y-type four-level system. We theoretically study the two- and three-photon coherence in this system. The results show that the coupling field makes the probing absorption intensity at the resonant frequency have a very narrow line-width depression, i.e., EIT. The microwave field causes a dynamic Stark splitting of the energy level |3> and induces the Aulter-Townes double peaks. Their frequency interval is exactly equal to the Rabi frequency of the microwave field. The presence of all three fields induces wide window of EIT at the line center owing to the enhanced depression results. The transient evolution is also discussed to understand the optical switching process in the system. Our theoretical study will be helpful to get a deeper insight into the three-photon effects in multilevel systems.
The question is raised whether nonlinear coherence can be created in coupled systems whose subsystems interact with distinct radiation fields. The simplest coupled system consisting of two two-level subsystems is treated in interaction with two electromagnetic fields, each close to resonance of one of the subsystems. The problem is treated within the SU(4) symmetries. The generators of SU(4) are defined and their properties under transformations deduced. The Hamiltonian and the density operator of the coupled system are expressed in terms of the generators of SU(4) and the Liouville equation is then used to find the time evolution of the density operator. When the frequencies of the radiation fields are equal to the resonant frequencies of the uncoupled subsystems, one obtains two two-photon nutation processes, provided that the interactions of the subsystems with the radiation fields are weaker than the mutual interaction of the two subsystems. The nutation frequency is proportional to the product of the field strengths and inversely proportional to the coupling parameter which in this case represents the resonance offset of the intermediate levels. Alternatively, the frequencies of the radiation fields can be arranged so that only one two-photon transition is on resonance. The amplitude of the nutation then depends on the strengths of the radiation fields, and acquires maximum value for ${\ensuremath{\omega}}_{1}={\ensuremath{\omega}}_{2}$. The nutation frequency increases when one approaches the resonance with the intermediate level. With a two-photon $\ensuremath{\pi}$ pulse one transfers the population difference from the levels of one subsystem to those of the other. This procedure can be used to polarize rare spins in the case of strong spin-spin coupling with the abundant spins when the method of thermodynamic contact is inapplicable. Both magnetic and optical coupled systems can be described with the same equations.
Electromagnetically induced transparency (EIT) resonance is obtained in a well-studied lambda three level system where an optical coupling field and a probing field interact with two optical transitions. However, both electromagnetically induced absorption (EIA) and EIT can be obtained in a modified quasi-lambda four level system consisting of an optical-radio two-photon coupling field and a probing field. By solving the density matrix equation of motion, a physical account of EIA and EIT is given in terms of a dressed state picture. It can be seen that the optical coupling field in this quasi-lambda four level system has a crucial effect on the forming of EIA and EIT. An EIA is observed under a resonant optical coupling and it evolves into an EIT when there is a detuning. Furthermore, the width of EIA or EIT increases as the optical coupling Rabi frequency increases.
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.