The Density Matrix and the Relaxation of Two-Level Systems

M. M. Al'perin, Ya. D. Klubis

Alexei Goun, Boris Ya Zel'dovich

Density matrix equations for the relaxation of a two-level system are studied. Standard dogma, Γ ik ≥ (γi + γk)/2, about the relaxation constant for the off-diagonal Γ ik ) and diagonal (γi) matrix elements is verified for the case of up- and down-transitions with high rates w 21 and w 12. Non-negativity of the density matrix imposes much weaker limitation, Γ21 ≥ (½)[(w 21)½ - (w 12)½]2. An almost ‘physical’ system is suggested, which describes to a certain approximation such decrease of transverse relaxation constant Γ.

Abraham Nitzan

Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.

Ya. D. Klubis

David E. Rourke, Larisa A. Khodarinova, Alexander A. Karabanov

A method is presented for the coherent control of two-level systems when T2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, Gamma, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T2. In general, the effect of T2 on Gamma cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.

J. Budimir