Two-Level System

A two-level atomic system subject to an optical field is analyzed.
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Time-Dependent Solution and the Rabi Frequency

We define an atomic system consisting of two states (a ground state labeled 1 and upper state labeled 2). This is a "toy" system that neglects angular momentum (J and M are not defined). We apply a light field detuned from resonance by a frequency .
We first neglect relaxation and solve for the dynamics of the system.
Define the atomic system.
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Here is what the density matrix looks like.
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Define the optical field with frequency . This is an x-polarized field by default, but for a toy system the polarization does not matter.
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The Hamiltonian for the system subject to the optical field.
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For simplicity of notation, we define the Rabi frequency R=(1||d1||2) E0.
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Apply the rotating-wave approximation to the Hamiltonian. We define the detuning =-Energy[2].
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Here are the evolution equations.
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Here are initial conditions, putting all the population in the ground state at t=0.
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Here is the solution.
Solve the system of equations with DSolve. We see that the upper state has population 2, 2(t) given by with generalized Rabi frequency and amplitude A=R2/(R2+2).
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Here is a plot of the populations.
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Steady-State Solution and the Saturation Parameter

If we add relaxation to the system, we can find a steady-state solution.
First we remove the time dependence from the density-matrix-element notation. This will have the effect of setting time derivatives of the density-matrix elements to zero.
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The density matrix with no explicit time dependence.
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There are two forms of relaxation present: intrinsic relaxation of the upper state due to spontaneous decay and "transit" relaxation due to atoms leaving the system. A "relaxation matrix" accounts for these processes.
IntrinsicRelaxation and TransitRelaxation supply the relaxation matrices.
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The ground state is repopulated by the same two mechanisms: spontaneous decay transfers atoms from the upper state to the ground state, and ground-state atoms enter the system. This is taken into account by the "repopulation matrix":
OpticalRepopulation and TransitRepopulation supply the repopulation matrices.
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Here are the evolution equations, including relaxation and repopulation. Since the density matrix elements are now explicitly time independent, we obtain time-independent equations for the steady state.
Create evolution equations with LiouvilleEquation.
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Here are the solutions.
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We can simplify these solutions by assuming that the transit rate is much slower than the natural width of the upper state, and by writing them in terms of the upper-state saturation parameter .
Set equal to zero and substitute for R.
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We see that the upper-state population is a power-broadened Lorentzian in detuning. Here is a plot of the upper-state population as a function of detuning for various values of the saturation parameter.
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Setting the detuning to zero, we see why is called the saturation parameter for this system: when is small, the upper-state population is linear in ; when is increased to unity, the upper-state population begins to saturate, and approaches its limiting value of 1/2.
The DM elements for resonant light in terms of the saturation parameter.
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The upper-state population as a function of .
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The saturation parameter also determines the width of the power-broadened Lorentzian.
Solve for the half-max value of detuning for the upper-state population.
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The full-width-half-max of the optical resonance.
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