In this video, we will discuss the time-dependence of a two-state system. Consider a two-level system or two-state system with a sinusoidal oscillating potential. The unperturbed Hamiltonian, H_0, is given as this. The energy level for state Number 1 is denoted as E_1. Energy level for state Number 2 is denoted as E_2. The time-dependent potential is given here. It has the sinusoidal dependence with an angular frequency Omega, amplitude is given as Gamma. Gamma and Omega are both assumed to be real and positive numbers. Since the potential here is given already in terms of the basis set of cap Number 1 and cap Number 2 against states of H_0, the matrix element that we need to construct for the time dependence is already given here. These quantities here simply are the off-diagonal elements. This expression does not contain any outer product of one and outer product of two. So the diagonal elements are zero, and so this is the equations that we need to solve to find the time-dependence of the coefficients c_1 and c_2. From that, we can obtain the time evolution of the quantum state. This matrix equation gives you a coupled equation for options c_1 and c_2, which we can uncouple by taking additional time derivative, and then, of course, the unknown constants are determined by the initial condition. Here, we assume that initially the system is in State 1 or energy Level 1 and Level 2 is empty. Solving that coupled equation yields this. The absolute value square of coefficient c_2 gives you the probability of finding the system in Level 2 or state Number 2. The probability of finding the system in Level 1 is simply given by 1 minus c_2 absolute value squared. The expression for c_2 absolute value square is given here, and here Omega21 is simply the energy difference between the two energy levels divided by h bar Omega. So Omega here, notice, is the oscillation frequency of your perturbation Hamiltonian and Omega 21 is related to the energy difference between the two energy levels of the given system. The probability of finding the system in Level 2 obviously oscillates. The frequency of oscillation is given here inside the sine squared function, and that frequency is explicitly written here and denoted as capital Omega. When the frequency of the potential Omega, here, is close to Omega 21, the energy difference between the two energy levels that minimizes the denominator of the factor in front of the sine square function. That maximizes the probability of finding the system in energy Level 2. So that maximizes the probability of making a transition from Level 1 to Level 2, and this condition is called the resonance condition. The frequency of the perturbation matches the energy difference of the system exactly, or the transition energy, transition probability, transition frequency exactly. Consider the case where the frequency of the perturbation potential is exactly equal to the frequency of transition, energy difference between the two energy level divided by h bar. Plugging in this condition into the equation for capital Omega in the previous slide will give you this simple factor of Gamma over h bar, and in this case, if we plot the quotients c_2 square and c_1 squared as a function of time, you will see these sine squared functions. So initially at t equals 0, the system was in Level 1. So c_1 square was at t equals 0 is 1, and c_2 squared at t equals 0 was 0. This was our initial condition, and as time evolves, there is this downward transition, the probability of finding the system in Level 1 decreases, and probability of finding the system in Level 2 increases. This period represents the absorption process. The system absorbs energy from the perturbation potential and promote itself into the higher energy level. Once it reaches this maximum point, it begins to decrease again. So the probability of finding the system in Level 2 decreases again, and probability of finding the system in Level 1 increases again. This is a period where the energy is released from the system and the system lowers itself in energy from Level 2 to Level 1, and this process just repeats itself indefinitely. Off resonance, where the frequency of the perturbation potential is not equal to the frequency of transition, then the oscillations still takes place. So sine squared function, which oscillates with the angular frequency of capital Omega still exist. However, the amplitude factor that's multiplied to the sine square function will have a different value, we will have a reduced value, smaller amplitude. If you look at the amplitude function, you will find that it has a form of a Lorentzian function, and the Lorentzian function, if you plot their amplitude as a function of Omega, the frequency of the perturbation potential, then it will peak, it will maximize at the resonance condition, and Omega equals Omega 21. As you detune from that resonance condition, the amplitude decreases according to the Lorentzian line shape. The full width at half maximum here is given by the four times Gamma over h bar. So it is related to the strength of your perturbation Hamiltonian. So when you have a strong perturbation Hamiltonian, then the width of this Lorentzian line shape is larger. So it will have a greater probability of making a transition even at off resonance conditions and vice versa. Here is an example that I found in the literature, and there is a molecule called DBATT, and it happens to have on energy level looking like this. There is a ground state and the excited state, and there is a transition between them, giving a fluorescence signal at roughly 589 nanometers. If you monitor these fluorescence signal under light illumination, which produces this oscillating perturbation potential, the time dependence of the fluorescence signal, as you can see, shows these oscillations, which is the oscillation that we just found out by solving these time-dependence for the two-level system.
