Atomic Density Matrix Tutorial | Tutorials »|More About » |

In[1]:= |

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 .

In[1]:= |

In[2]:= |

Out[2]//MatrixForm= | |

Define the optical field with frequency . This is an *x*-polarized field by default, but for a toy system the polarization does not matter.

In[3]:= |

Out[3]= |

In[4]:= |

Out[4]//MatrixForm= | |

In[5]:= |

Out[5]//MatrixForm= | |

In[6]:= |

Out[6]//MatrixForm= | |

In[7]:= |

Out[7]//TableForm= | |

In[8]:= |

Out[8]//TableForm= | |

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=_{R}^{2}/(_{R}^{2}+^{2}).

In[9]:= |

Out[9]//TableForm= | |

In[10]:= |

Out[10]= |

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.

In[11]:= |

Out[11]= |

In[12]:= |

Out[12]//MatrixForm= | |

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.

In[13]:= |

Out[13]//MatrixForm= | |

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":

In[14]:= |

Out[14]//MatrixForm= | |

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.

In[15]:= |

Out[15]//TableForm= | |

In[16]:= |

Out[16]//TableForm= | |

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 .

In[17]:= |

Out[17]//TableForm= | |

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.

In[18]:= |

Out[18]= |

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.

In[19]:= |

Out[19]= |

In[20]:= |

Out[20]= |

In[21]:= |

Out[21]= |

In[22]:= |

Out[22]= |