1. Bloch spheres, Bloch vectors, and solving Schrödinger’s equation with (almost) no math Two-level time independent hamiltonians David Blasing, Quantum Mechanics 12/07/2015

2. A general two-level time independent hamiltonian: H, is decomposable into H 0 + H p, where H 0 is the unpreturbed “free” hamiltonian and H p is the pertubring hamiltonian that “couples” the unperturbed eignstates << 1, we’ll later use this - - + H = - h H p = This lecture is about understanding the effect of H p …why should you care? + - - h - h h - h -

3. The Bloch sphere formalism efficiently helps explain many physical systems Here are 14 examples where physicists used Bloch spheres as visual tools…(there are many more)

4. Outline: 1.The Bloch sphere and the Bloch vector 2.A hamiltonian and its rotation vector 3.The rotating wave approximation and rotating frame transformation 4.The geometric pictures of the four physical parameters of interest

5. Bloch sphere and Bloch Vector Mathematically, a two-state wavefunction, Ψ = C e |e>+C g |g>, requries four real numbers. Not all four are needed physically: Normalization (|C e | 2 +|C g | 2 = 1) reduces the required number by one The overall phase doesn’t have a physical consequence From the remaining two, the Bloch sphere (BS) and the Bloch vector (BV) are built…

6. Outline: 1.The Bloch sphere and the Bloch vector 2.A hamiltonian and its rotation vector 3.The rotating wave approximation and rotating frame transformation 4.The geometric pictures of the four physical parameters of interest

7. An example hamiltonian and its rotation vector from NMR In bra-ket notation: In matrix form: The hamiltonian describing the spin/magnetic field interaction is: where and a typical magnetic field for an NMR experiment is

8. A hamiltonian and its rotation vector Make some definitions: and, note the units of radians per second (a rate of rotation) Given these, Written in matrix form: /2 +

9. Bloch sphere and Bloch Vector H changes C g and C e throughout time, so it moves BV on BS

10. Bloch sphere and Bloch Vector (, magnetic fields make BV anti-rotate around themselves…magnetic field along –Z cause a right-handed rotation around +Z.) So let’s see what Schrodinger’s equation gets us: cece cgcg ) = cece cgcg ) ( /2

11. all have different effects…only two are “important”

12. Outline: 1.The Bloch sphere and the Bloch vector 2.A hamiltonian and its rotation vector 3.The rotating wave approximation and rotating frame transformation 4.The geometric pictures of the four physical parameters of interest

13. Rotating wave approximation: Since Ω - rotates BV up and then down, it averages to 0 (when t>>1/ω 0 ) So RWA is:

14. Rotating frame transformation:

15. Mathematically, transforming to the rotating frame is a “unitary transformation.” Instead of |e> and |g>, we now use e (-iωt/2) |e> and e (+iωt/2) |g> as our basis. After this transformation, the hamiltonian in the new basis is: - + H = - h which we label as - + - h /2

16. Rotating frame transformation: ( =) Passing note:

17. Rotating frame transformation: The hamiltonian determines the rotation vector The initial quantum state determines the initial BV The hamiltonian and it’s interaction time with the quantum state determine the angle of rotation Final BV is determined by the initial quantum state, the hamiltonian and the interaction time

18. Outline: 1.The Bloch sphere and the Bloch vector 2.A hamiltonian and its rotation vector 3.The rotating wave approximation and rotating frame transformation 4.The geometric pictures of the four physical parameters of interest

19. Four physical parameters of interest: