Two-Level Systems: Coupling and Evolution

1. Chapter 4 Two-Level Systems, Spin

2. 4.C.1 Two-level systems • Let us start with the simplest non-trivial state space, with only two dimensions • Despite its simplicity, such space is a good approximation of many physical quantum systems, where all other energy levels could be ignored • If the Hamiltonian of the system is H0, then eigenvalue problem can be written as:

3. 4.C.1 4.C.2 Coupling in a two-level system • To account for either external perturbations or the neglected internal interactions of the two-level system, an additional (small) inter-level coupling term is introduced in the Hamiltonian: • In the original (unperturbed) basis the matrix of the perturbed Hamiltonian can be written as: • Let us assume that the coupling is time-independent • Since the coupling perturbation is observable

4. 4.C.1 4.C.2 Coupling in a two-level system • What modifications of the two-level system will such coupling introduce? • Now, the eigenvalue problem is modified: • Thereby one has to find the following relationships: • In other words, the new (perturbed) eigen-problem has to be diagonalized

5. 4.C.2 Coupling in a two-level system • The solution is:

6. 4.C.2 Coupling in a two-level system • The solution is:

7. 4.C.2 Coupling in a two-level system • The solution is:

8. 4.C.2 Coupling in a two-level system • The solution is:

9. 4.C.3 Evolution of the state vector • Let at instant t the state vector is a superposition of the two “uncoupled” eigenvectors: • Since • we get:

10. 4.C.3 Evolution of the state vector • On the other hand: • Recall that if • then • Thus, assuming

11. 4.C.3 Evolution of the state vector • On the other hand: • Recall that if • then • Thus, assuming • one gets

12. 4.C.3 Evolution of the state vector • Let us choose a special case: • Recall that • Then • Since

13. 4.C.3 Evolution of the state vector • The probability amplitude of finding the system at time t in state :

14. 4.C.3 Evolution of the state vector • The probability amplitude of finding the system at time t in state : • Then the probability is • The system oscillates between two “unperturbed” states

15. 4.C.2 Applications: quantum resonance • If • then the unperturbed Hamiltonian is 2-fold degenerate • The inter-level coupling lifts this degeneracy giving rise to the ground state and an excited state • E.g., the benzene molecule • It has two equivalent electronic states

16. 4.C.2 Applications: quantum resonance • However, there is coupling between the two states, so that the perturbed Hamiltonian matrix has non-diagonal elements • The two levels become separated • This makes the molecule more stable since the ground state energy is below Em while the ground state is a resonant superposition of the unperturbed states

17. 4.C.2 Applications: quantum resonance • Another example: a singly ionized hydrogen molecule • There is coupling between two equivalent electronic states, yielding a lower ground state energy • This leads to the delocalization of the electron – the ground state is a resonant superposition of the unperturbed states, which is in essence a chemical bond

18. 4.A.2 Applications: spin • Let us label the elements of the perturbed Hamiltonian matrix as follows:

19. 4.A.2 Applications: spin • The perturbed Hamiltonian matrix • can be rewritten introducing a matrix corresponding to a vector operator • This matrix is Hermitian, therefore it should correspond to some observable • Let’s define this observable as the spin • Then

20. 4.A.2 Applications: spin • It turns out that such observable indeed exists • Moreover, it is one of the most fundamental properties of an electron and other elementary particles • Spin can be measured experientially, and it gives rise to many macroscopic phenomena (such as, e.g., magnetism)

21. 4.B.3 Some properties of spin operator • Let’s consider a uniform field B0 and chose the direction of the z axis along the direction of B0 • Within the notation • So, the eigenvalue problem for H is: • The two states correspond to the spin vector parallel and antiparallel to the field

22. 4.B.3 Some properties of spin operator • We can apply the formulas we derived for the two-level system • For example: • This is precession