Question

1. The aim of this problem set is to understand the dynamics of a spin-1/2 system in its full glory. Note that formally a spin-1/2 system and a qubit are equivalent hence, all what you will discover in this problem set will carry over to single qubits. Consider an electron spin (spin 1/2, magnetic moment gHB) interacting with a strong magnetic field Bo (0,0, B) in the z direction as well as with a much weaker magnetic field Brf = (B0 cos arft, B0sinwrft. O) rotating in the x, y plane with speed wrf. We are interested in understanding the spin's time evolution and how it can be manipulated (a) What is the corresponding Hamiltonian and Schrödinger equation in its ex point (b) At this point, we could solve the differential equation directly, however, to build intuition and to help with future more complicated scenarios, we will now use a trick and transform into a rotating frame such that the Hamil- tonian becomes time-independent. The transformation is motivated by the observation that for classically evolving magnetic moment we could apply a basis transformation such that the rotating frame magnetic field becomes plicit matrix formulation in the l ±z) basis? stationary. That is we would expect that the spin of an electron in the |+a state whose spin points in the direction of the rotating magnetic field Brf at t = 0 would always be aligned with Br simplifying the problem consider- First we are going to understand how the Schrödinger equation changes un- der a general basis transformation. Consider a general basis transformation such that l Ψ),ot-U )lab with U describing the basis transformation from the laboratory frame to the rotating frame. Show that in the new basis, we can rewrite the Schrödinger equation in the same form just with a modified 1 point Hamiltonian Hrot-UHınd

Answer 1

Spin half particle H = -mu^rightarrow B^rightarrow = -gamma s^rightarrow B^rightarrow gamma = g gamma/2m g = gy r_o magnetic ratio = -gamma h/2 laft rightarrow B^rightarrow H = n sigma^rightarrow B^rightarrow [lef A = -r h/2] B^_0 rightarrow_rightarrow (0, 0, B_2) B_yc = (B_0 correct, B_0 sin wt, 0) H = A(sigma_A B_x + sigma_y B_y + sigma_2 B_2) = A { B_0 cos wt (0 1 1 0) + B_0 sin wt (0 I - I 0) + B - z (1 0 0 - 1) = A(Bz B_0 B_0 cos t we - i B_0 sin wt cos wt + 2 degree B_0 sin wt - Bz) = A(Bz B_0 l^iwt B_0 l^- iwt_- Bz) now solving the matrix equation def (ABZ - lambda AB_0 o^iwt AB_0 l^- iwt - ABZ) = 0 - (A^2 B z^2 - lambad^2) = A^2 B^2_0 lambda = plusminus