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arXiv:quant-ph/0504065 · Journal of Physics A Mathematical and General

Quadratic pseudosupersymmetry in two-level systems

Boris F Samsonov, V V Shamshutdinova

Published May 10, 2005• 11 citations
arXivHTMLPDF

Abstract

Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time-dependent electric fields for which the equation of motion for a two-level system placed in this field can be solved exactly is obtained. A new interesting phenomenon is observed. There exists a time-dependent detuning of the field frequency from the resonance value such that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to 3/4. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with small amplitude and frequency close to the Rabi frequency (fast oscillations) take place at the background of those with big amplitude and small frequency (slow oscillations). During the period of slow oscillations, the minimal value of the probability to populate the excited level may exceed 1/2, suggesting for an ensemble of such two-level atoms the possibility of acquiring an inverse population and exhibit lasing properties.

DOI
10.1088/0305-4470/38/21/015

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Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time-dependent electric fields for which the equation of motion for a two-level system placed in this field can be solved exactly is obtained. A new interesting phenomenon is observed. There exists a time-dependent detuning of the field frequency from the resonance value such that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to 3/4. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with small amplitude and frequency close to the Rabi frequency (fast oscillations) take place at the background of those with big amplitude and small frequency (slow oscillations). During the period of slow oscillations, the minimal value of the probability to populate the excited level may exceed 1/2, suggesting for an ensemble of such two-level atoms the possibility of acquiring an inverse population and exhibit lasing properties.
Abstract. Using the intertwining relation we construct a pseudosuperpartner for a (non-Hermitian) Diraclike Hamiltonian describing a two-level system interacting in the rotating wave approximation with the electric component of an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry generators close a quadratic pseudosuperalgebra. A class of time dependent electric fields for which the equation of motion for a two level system placed in this field can be solved exactly is obtained. New interesting phenomenon is observed. There exists such a time-dependent detuning of the field frequency from the resonance value that the probability to populate the excited level ceases to oscillate and becomes a monotonically growing function of time tending to 3/4. It is shown that near this fixed excitation regime the probability exhibits two kinds of oscillations. The oscillations with a small amplitude and a frequency close to the Rabi frequency (fast oscillations) take place at the background of the ones with a big amplitude and a small frequency (slow oscillations). During the period of slow oscillations the minimal value of the probability to populate the excited level may exceed 1/2 suggesting for an ensemble of such two-level atoms the possibility to acquire the inverse population and exhibit lasing properties.
The problem of a single two level atom in its excited state, placed in a cavity in which no photons are present initially, and interacting with N modes of the electromagnetic field is solved exactly in the electric dipole, rotating wave approximation. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time dependent wavefunction of the system is given. These quantities are evaluated analytically for N=1,2 and 3, and the time dependent properties of the system for these values of N are investigated and discussed.
It is shown that the population Rabi oscillations in a lossless two-level atom, interacting with a monochromatic electromagnetic field, in general, are convergent in time. The well-known continuous oscillations take place because the restricted choosing of initial conditions, that is, when the atom is chosen on ground or excited level before the interaction, simultaneously having a definite value of momentum there. The convergence of Rabi oscillations in atomic wave-packet states is a direct consequence of the Doppler effect on optical transition rates (Rabi frequencies): it gradually leads to ``irregular'' chaotic-type distributions of momentum in ground and excited energy levels, smearing the amplitudes of Rabi oscillations. Conjointly with Rabi oscillations, the coherent accumulation of momentum on each internal energy level monotonically diminishes too.
We considered the semiclassical Rabi model: a two-level atom fixed at a position and interacting with a classical, monochromatic electromagnetic field. In resonance and for an arbitrary initial state of the system, we obtained an approximate, analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/21/21/2 (1/41/41/4) of the angular frequency of the field and 111 (101010) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA. We found that the approximate solution is composed of the the solution in the RWA plus terms that oscillate at the angular frequency of the field and at the Rabi frequency. Moreover, these corrections are proportional to the Rabi frequency divided by two times the angular frequency of the field.
Chains of first-order SUSY transformations for the spin equation are studied in detail. It is shown that the transformation chains are related with a olynomial pseudo-supersymmetry of the system. Simple determinant formulas for the final Hamiltonian of a chain and for solutions of the spin equation are derived. Applications are intended for a two-level atom in an electromagnetic field with a possible time-dependence of the field frequency. For a specific form of this dependence, the time oscillations of the probability to populate the excited level disappear. Under certain conditions this probability becomes a function tending monotonously to a constant value which can exceed 1/2.
Rabi oscillations of a two-level atomic system with damping effects are described by the use of a pseudo-Hermitian Hamiltonian. The Rabi frequency and the amplitudes for being in the upper and lower level depend in the present system on both the Hermitian resonant interaction between the em field and the atomic dipole and on damping effects. The time development of the two-level system is studied by using the metric of the pseudo-Hermitian Hamiltonian and its bi-orthonormal basis of states. The special characteristics of the present system are related to PT (parity-time reversal) and C (generalized conjugation operator) invariant properties of the pseudo-Hermitian Hamiltonian.