Here we explore how coupling to a continuum of states leads to irrevesible relaxation out of an initial state.
We will calculate the time-evolution of amplitude in an initially prepared state for a finite number of continuum states. The number of continuum states can be adjusted and we see how the occupation of the initial states evolves from purely oscillatory to exponentially damped as the number of states is increased from 1 to 1000.
Set some variables:
Coupling to continuum states:
Define a time grid:
k=state index variable:
k=0 is intial state
Rabi frequency:
Choose the continuum state energies:
Plot spectrum of continuum density of states:
For flat continuum:
Amplitude in initial and continuum state.
Intitial state:
our assumptions wont conserve population, so we'll fix the absolute numbers with it with:
Now let's compare the results to what you get from the Golden rule rate:
Density of states:
Average density of states:
It's more accurate to use delta function!
...or input your own number
Golden rule rate: