C/CS/Phys 191 Spin operators, spin measurement, spin initialization 10/13/05 Fall 2005 Lecture 14. = state representing ang. mom. w/ z-comp.

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1 C/CS/Phys 9 Spin operators, spin measurement, spin initialization //5 Fall 5 Lecture 4 Readings Liboff, Introductory Quantum Mechanics, Ch. Spin Operators Last time: + state representing ang. mom. w/ z-comp. up state representing ang. mom. w/ z-comp. down These are the eigenvectors and eigenvalues of the spin for a spin- and are simultaneous eigenvectors of S and S z. system, like an electron or proton: S S S z S z h s(s+ h ( + 4 h h s(s+ 4 h hm h,m + hm h,m Results of measurements: S 4 h,s z + h, h Since S z is a Hamiltonian operator, and form an orthonormal basis that spans the spin- space, which is isomorphic to C. So the most general spin state is ( Ψ α + β α. β Question: How do we represent the spin operators (S,S x,s y,s z in the -d basis of the S z eigenstates and? Answer: They are matrices. Since they act on a two-dimensional vectors space, they must be -d matrices. We must calculate their matrix elements: S ( s s s s ( sz s,s z z s z s z ( sx s,s x x s x s x, etc. (S y C/CS/Phys 9, Fall 5, Lecture 4

2 Calculate S matrix: We must sandwich S between all possible combinations of basis vectors. (This is the usual way to construct a matrix! s S 4 h 4 h s S 4 h s S 4 h s S 4 h 4 h So S 4 h ( Find the S z matrix: So S z h s z Sz h + h s z S z h s z Sz h + s z Sz h h ( Find S x matrix: This is more difficult What is S x S x? is not an eigenstate of S z, so it s not trivial. Use raising and lowering operators: S ± S x ± is y S x (S + + S,S y i (S + S S x (S + + S S+, since is the highest Sz state. But what is S? Since S is the lowering operator, we know that S. That is S A for some complex number A which we have yet to determine. Similarly, S + A+. Question: What is A? Answer: A + h s(s+ m(m+ S + s,m A+ s,m+ A h s(s+ m(m S s,m A s,m Proof: First note that A s,m S+ S s,m. C/CS/Phys 9, Fall 5, Lecture 4

3 Since S ± S x ± is y, where S x, S y are Hermitian, S + S. Hence A + (s,m A (s,m+. We are free to choose a phase for the eigenstates; set it so A + (real and nonnegative. Now Also, s,m S S + s,m (S s,m S + s,m (S+ s,m S + s,m A + s,m s,m A +. S S + (S x is y (S x + is y S x + S y + i[s x,s y ] S S z hs z. Therefore, s,m S S + s,m s,m (S S z hs z s,m h s(s+ ( hm h( hm h (s(s+ m(m+, using S s,m h s(s+ s,m and Sz s,m hm s,m. Thus A + (s,m h s(s+ m(m+ A (s,m A + (s,m h s(s+ m(m. Now we use these values A ± to find the desired coefficients: S + S + S S h ( + ( ( + h h ( + ( ( h S x (S+ + S [S+ + S ] S x [+ h ] So S x h S x (S + + S [ h + ] h S x (S + + S [+ h h ] S x (S + + S [ h + ] ( C/CS/Phys 9, Fall 5, Lecture 4

4 Find S y matrix: Use S y i (S + S. The proceed similarly to above for S x. Check it out yourself to see that you understand the matrix and bra-ket mechanics. ( i Answer: S y h i In summary, we define ( ( σ σ ( i σ i ( σ Then, S 4 h σ, S x h σ, S y h σ, S z h σ σ,σ,σ,σ are called the Pauli Spin Matrices. They are very important for understanding the behavior of two-level systems. Note that we have already encountered these in our discussion of qubits, where they correspond to the gates X σ, Y σ, Z σ, I σ. In the next couple of lectures we shall frequently interconvert, using S x h X, S y h Y, S z h Z. Measuring Spin We can measure the spin state m with a Stern-Gerlach device. This is simply a magnet set up to generate a particular inhomogeneous B field. In the figure below, the field is strong near the N pole (below and weaker near the S pole (above. If the z-axis points upwards, then we have a negative field gradient in the z direction, i.e., B/ z <. µ S N up down When a particle with spin state ψ α + β is shot through the apparatus from the left, its spin-up portion is deflected upward, and its spin-down portion downward. The particle s spin becomes entangled with its position! Placing detectors to intercept the outgoing paths therefore measures the particle s spin. Why does this work? We ll give a semiclassical explanation mixing the classical F m a and the quantum Hψ Eψ which is not really correct, but gives the correct intuition. [See Griffith s 4.4., pp for a more complete argument.] Now the potential energy due to the spin interacting with the field is E µ B, so the associated force is F spin E ( µ B. C/CS/Phys 9, Fall 5, Lecture 4 4

5 At the center B B(zẑ, with B z <, so F B (µ z B(z µ z z ẑ. The magnetic moment µ is related to spin S by µ gq ms e ms for an electron. Hence F e m B z S zẑ ; if the electron is spin up (S z +/, the force is upward, and if the electron is spin down (S z /, the force is downward. 4 Initialize a Spin Qubit How can we create a beam of qubits in the state ψ? Pass a beam of spin- particles with randomly oriented spins through a Stern-Gerlach apparatus oriented along the z axis. Block the downward-pointing beam, leaving the other beam of qubits. Note that we measure the spin when we intercept an outgoing beam measurement process is probabilistic and not unitary. How can we create a beam of qubits in the state ψ α +β? First find the point on the Bloch sphere corresponding to ψ. That is, write ψ α + β cos θ + e iϕ sin θ (up to a phase, where tan θ β α eiϕ β/ β α/ α. The polar coordinates θ,ϕ determine a unit vector ˆn cosϕ sin θ ˆx+sinϕ sinθŷ+cosθẑ. Now just point the Stern-Gerlach device in the corresponding direction on the Bloch sphere, and intercept one of the two outgoing beams. That is, a Stern-Gerlach device pointed in direction ˆn measures S ˆn ˆn S. ˆ Some details for this. We need to find the eigenstates of S n S ˆn. First express this in cartesian coordinates, using ˆn sinθ cos φ ˆx+sinθ cosφŷ+cosθẑ, and the relations S α h/α,α X,Y,Z. We thereby arrive at the x matrix representation of S n in the z-basis: S n h ( cos θ sin θe iφ sinθe iφ, cosθ Now diagonalize this to obtain eigenvalues ± h/ (why are you not surprised? and eigenstates n cos θ + e iϕ sin θ (s n + h n e iθ sin θ + cos θ (sn h. The first eigenstate is the desired general state on the Bloch sphere, so we just need to intercept the positive spin eigenstate, i.e., upward deflected beam in the S-G frame. Can we use an S-G apparatus to implement a unitary (deterministic transformation? No, since the S-G apparatus makes a measurement and is not unitary. In other words, it collapses the wave function to one component only. It is fine for initializing an arbitrary state, but to make a unitary transformation we have to evolve the wave function according to a Hamiltonian Ĥ: ψ(t e ī h Ĥt ψ( C/CS/Phys 9, Fall 5, Lecture 4 5

6 This rotates the spin qubit wave function on the Bloch sphere. In the next lecture we will see how to accomplish an arbitrary single-qubit unitary gate (a arbitrary rotation on the Bloch sphere by two different approaches, Larmor precession and spin resonance. C/CS/Phys 9, Fall 5, Lecture 4 6