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Two-Level Systems by Hand — Timing a π-pulse

Exercises

Introductory Quantum Mechanics › Unit 3 · Two-level systems & spin-1/2 › Two-Level Systems by Hand › all problems

Practice Problem 7 of 9

Timing a π-pulse

Problem

In an NMR experiment the rotating transverse field is

B_{1} = 10 μ

T. Using the proton

γ

from the Larmor problem, how long must the pulse last to invert the spin (

π

-pulse)? What pulse duration gives an equal superposition (

π /2

-pulse)?

Solution

Predict before reading on. On resonance the Rabi formula is complete transfer at $Ω_{1} t = π$ with $Ω_{1} = γ B_{1}$ . Which quantity from the worked example does $γ B_{1}$ replace?

f_{1} = \frac{γ B _{1}}{2 π} = 425.8 Hz \Rightarrow t_{π} = \frac{1}{2 f _{1}} = 1.17 ms, t_{π /2} = 0.59 ms

Millisecond pulses, kilohertz Rabi frequencies, megahertz Larmor frequencies: three well-separated scales, which is what makes the rotating-wave picture excellent here. Every pulse sequence in NMR and every single-qubit gate is built from these two durations.

Result

t_{π} = 1.17 ms, t_{π /2} = 0.59 ms