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Unformatted text preview: C/CS/Phys 191 Spin manipulation II (resonance), quantum gates on spins 10/20/05 Fall 2005 Lecture 16 1 Readings Liboff, Introductory Quantum Mechanics, Ch. 11 Stolze and Suter, Quantum Computing, Ch. 10 Nielsen and Chuang, Quantum Computation and Quantum Information, Ch. 7.7.2, 7.7.3 2 Spin Resonance and single qubit gates How do we control qubit states in the lab? If vextendsingle vextendsingle ψ ( t ) )big = α ( t ) vextendsingle vextendsingle )big + β ( t ) vextendsingle vextendsingle 1 )big , how do we deterministically change α and β ? We know that the Hamiltonian evolves things in time, so if we turn on a field then the Hamiltonian will evolve the state via e- i ˆ Ht / ¯ h . For a static magnetic field we saw in the last lecture that this allows us to rotate qubit state from one point on the Bloch sphere to another via Larmor precession: ˆ R i ( ∆ θ ) = e- i ˆ S i ∆ θ / ¯ h , ∆ θ = eB o m ∆ t , vector B = B o ˆ x i This rotation has to occur at a rate determined by the magnitude of B which is fast. To get better control we would like to have a slower rotation. Question: How can we maintain energy level splitting between vextendsingle vextendsingle )big and vextendsingle vextendsingle 1 )big and control the rate at which a qubit rotates between states? (i.e. change it at a rate different from ϖ o = eB o m .) Answer: Spin Resonance gives us a new level of control (most clearly seen in NMR). How it works: Turn on a big DC field B o and a little AC field vector B sin( ϖ o t) that is tuned to the resonance ϖ o = eB o m : Figure 1: The small AC field induces controlled mixing between vextendsingle vextendsingle )big and vextendsingle vextendsingle 1 )big ... “SPIN FLIPS”. We must solve the Schrodinger equation to understand what is going on: i ¯ h ∂ ∂ t vextendsingle vextendsingle ψ ( t ) )big = ˆ H vextendsingle vextendsingle ψ ( t ) )big C/CS/Phys 191, Fall 2005, Lecture 16 1 It is convenient to use column vector notation: vextendsingle vextendsingle ψ ( t ) )big = α ( t ) vextendsingle vextendsingle )big + β ( t ) vextendsingle vextendsingle 1 )big = parenleftbigg α ( t ) β ( t ) parenrightbigg What’s the Hamiltonian? ˆ H =- vector μ · vector B = e m vector S · vector B We now let the magnetic field be composed of the large bias field as before, together with a small oscillating transverse field: vector B = B o ˆ z + B 1 cos ϖ o t ˆ x With this we obtain the Hamiltonian: ˆ H = e m B o ˆ S z + e m B 1 cos ϖ o t ˆ S x Now use 2 × 2 matrix formulation, where the Pauli matrices ( ˆ S z = ¯ h 2 σ z , etc.) are of course eminently useful: ˆ H = e m B o · ¯ h 2 parenleftbigg 1- 1 parenrightbigg + e m B 1 cos ϖ o t · ¯ h 2 parenleftbigg 1 1 parenrightbigg The two terms sum to give the following 2 × 2 Hamiltonian matrix (expressed in the ˆ S z basis): ˆ H = e ¯ h 2 m parenleftbigg B o B 1 cos ϖ o t B 1 cos ϖ o t- B o parenrightbigg Now we can plug this Hamiltonian into the Schr. equation and solve forNow we can plug this Hamiltonian into the Schr....
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- Fall '05
- mechanics, Bloch sphere, spin resonance