English: The energy levels of the antihydrogen in the n = 1 and n = 2 states are depicted as a function of the magnetic field. On the vertical axis, the centroid energy difference, E1S–2S = 2.4661 × 1015 Hz, has been suppressed. The dotted vertical black line represents the field at the magnetic minimum of our trap, 1.0327 T (see above). Details of the energy levels near this field and their state labels are shown on the right of the figure. The first value in the ket notation represents the quantum number of the projection of the total angular momentum of the positron, mL + mS, where L is the orbital angular momentum (L = 0 for the S state and L = 1 for the P state, respectively) and S is the spin (S = 1/2). The double arrow shows the antiproton spin (up or down). Initially, both the 1Sc and 1Sd states are trapped in our magnetic trap. The grey arrow indicates the microwave-driven 1Sc → 1Sb transition to eliminate the anti-atoms in the 1Sc hyperfine state and prepare a doubly spin-polarized antihydrogen sample in the 1Sd state. The solid and broken red (cyan) arrows indicate the cycling transition for laser cooling (heating) with red (blue) detuning −δ (+δ′). The purple arrow represents the probe laser excitation to the 2Pc– level. Note that the 2Pc state at a magnetic field of about 1 T is a superposition of the positron spin-up (mL = 0, mS = +1/2) and spin-down (mL = +1, mS = –1/2) states. Owing to this superposition, upon de-excitation from the 2Pc state, the anti-atom can either go back to the original 1Sd state, or undergo an effective ‘spin flip’ transition to the 1Sa state. In the latter case, the anti-atom is forced out of the trap and detected via its annihilation signal. The black arrows show the two-photon excitation from the 1Sd state to the 2Sd state.