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Modern Physics Letters B

A CLOSED-FORM SOLUTION OF THE EVOLUTION OPERATOR IN TWO-LEVEL SYSTEMS

Xian-Geng Zhao

Published Apr 20, 1994• 1 citation
Publisher

Abstract

It is demonstrated by using the technique of Lie algebra SU(2) that the problem of two-level systems described by arbitrary time-dependent Hamiltonians can be solved exactly. A closed-form solution of the evolution operator is presented, from which the results for any special case can be deduced.
DOI
10.1142/s0217984994000534

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We construct an explicit Lewis-Ermakov-type dynamical invariant for time-dependent two-level systems by exploiting their algebraic correspondence with the time-dependent harmonic oscillator through the su(2)su(2)su(2) and su(1,1)su(1,1)su(1,1) algebras, which share the common complexification sl(2,C)sl(2,\mathbb{C})sl(2,C). This invariant provides a closed-form evolution operator and an exact propagator for arbitrary time-dependent coupling a(t)a(t)a(t) and detuning b(t)b(t)b(t). We illustrate the method on representative scenarios, including Landau-Zener transitions, non-Hermitian dissipative processes, and adiabatic rapid passage, and we obtain inverse-engineered shortcuts to adiabaticity with fully explicit control fields when the auxiliary scaling function is taken to be real.