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Physical Review A

Nonlinear magnetic and optical coherence in coupled two-level systems

Antonije Dulcic

Published Dec 1, 1983• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.28.3467

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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 \ensuremathω1=\ensuremathω2{\ensuremath{\omega}}_{1}={\ensuremath{\omega}}_{2}\ensuremathω1​=\ensuremathω2​. The nutation frequency increases when one approaches the resonance with the intermediate level. With a two-photon \ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
The problem of nonlinear coherence in three-level systems exposed to two monochromatic radiation fields is treated using the unitary symmetries of SU(3). The Hamiltonian and the initial density operator are brought by a series of unitary transformations into a form in which the problem is reduced to a nutation between the two levels connected by the two-photon transition, i.e., to a time-dependent unitary rotation within the proper SU(2) subgroup. The net effect of the two modes in which the photons can combine in the nonlinear transition is shown to vanish gradually as the three levels approach an equidistant pattern. This certifies that two-photon processes can occur in systems which obey SU(3) symmetry, but not in those which obey the symmetry of SU(2). Four independent constants of motion are identified for the process of two-photon nutation in three-level systems.
The time evolution of a two-level system interacting with a classical monochromatic radiation field is calculated by Fourier transforming the equations of motion. This can be done exactly if one uses the convolution of the product of two Fourier transformations. The method produces recurrence relationships for the transforms. The time-dependent amplitudes of the states and the transition probability are evaluated by using two such recurrence relationships. The time-averaged probability shows characteristic resonances at odd multiples of the ratio of the natural frequency of the system to the frequency of the radiation field. The peaks of the resonances are displaced by the Bloch-Siegert shift. At these resonances the system is completely saturated with the flipping frequency dependent on the strength of the radiation field. The time dependence and the frequency spectrum of the dipole moment is determined at the main resonance and at the three-quantum resonance. The spectrum contains the zero-frequency terms which are responsible for the coherence resonances.
In recent years interest has significantly quickened in investigation of nonlinear optical and spectroscopic phenomena in such unique objects as Rydberg atoms (the atoms in the energy states near the ionization limit) and van der Waals complexes (dimers). These objects are characterized, in particular, by that their energy states are strongly degenerate due to a large value of angular (rotational) momentum. Here, the problem arises of adequate theoretical description of experimental results, especially when polarization peculiarities of nonlinear radiation processes are considered. The problem is caused by the existence of a huge number of quantum magnetic sublevels of energy states, which are, in the general case, coherently coupled to each other and take part in the radiation processes. Figure 1 gives a vivid presentation of the problem arising even in such a seemingly simple case when one need describe the radiation transitions caused by a strong monochromatic field between two energy levels with large values of the angular momentum J. If the field polarization is arbitrary, the whole set of the magnetic sublevels turn out to be coupled by radiative transitions. Formally, the problem becomes essentially multilevel. In this case not only the hope fades of solving it analytically using the traditional quantum-mechanical approach, but the problems of numerical solution grow abruptly with increasing quantum number J as well. On the other hand, as is often the case, every cloud has a silver lining. With large J we approach the situation when the rotational motion of particles can be described classically. It turns out that the transition from the quantum description of orientation states of the angular momentum to the classical one simplifies the problem radically: it reduces to the problem of radiation processes in the simplest model of nondegenerate states. In this case the rotational motion manifests itself in a dependence of the density matrix elements on orientation angles of the angular momentum as on a parameter.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.
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 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 $\ensuremath{\pi}$ 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.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by rNT\frac{\mathrm{rN}}{T}TrN​, where NNN is the total number of atoms, TTT is the average time of coherent interaction of atoms and the radiation field, and rrr is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.