conventions and notation


 Rudolf Osborne
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1 Ph95a lecture notes, //0 The Bloch Equations A quick review of spin conventions and notation The quantum state of a spin particle is represented by a vector in a twodimensional complex Hilbert space H. Let z è and z è be eigenstates of the operator corresponding to component of spin along the z coordinate axis, S z z è z è, S z z è z è. In this basis, the operators corresponding to spin components projected along the z,y,x coordinate axes may be represented by the following matrices: S z z 0 0, S x x 0 0, S y y 0 i. i 0 We commonly use x è to denote eigenstates of S x, and similarly for S y. The dimensionless matrices x, y, z are known as the Pauli matrices, and satisfy the following commutation relations: x, y i z, y, z i x, z, x i y. 3 In addition, Tr i j ij, 4 and x y z. 5 The Pauli matrices are both Hermitian and unitary. An arbitrary state for an isolated spin particle may be written è cos exp i z è sin exp i z è, 6 where and are real parameters that may be chosen in the ranges 0 â â and 0 â â. Note that there should really be four real degrees of freedom for a vector in a
2 twodimensional complex Hilbert space, but one is removed by normalization and another because we don t care about the overall phase of the state of an isolated quantum system. Through the, representation, an arbitrary pure state may be represented as a point on the surface of a sphere, often referred to as the Bloch Sphere, with as polar angle (latitude) and as azimuthal angle (longitude). The vector pointing from the original (center) of the Bloch Sphere to the point representing a quantum state is known as the Bloch vector corresponding to that state. The north and south poles thus correspond to z è and z è, respectively, and x è Û, 0, x è Û,, y è Û,, y è Û, 3. 7 With the sign conventions we have chosen, directions on the Bloch Sphere correspond to directions in coordinate space. Hence the state corresponding to spin pointing along a unit vector û u x u y u z 8 (with u x u y u z ) has Bloch angles sin u x u y, tan u y u x. 9 Likewise, the operator corresponding to component of spin along the ûdirection is S u S û sin cos S x sin sin S y cos S z Û cos sin exp i sin exp i cos. 0 The, representation defined above has the additional nice property that
3 ès x è cos exp i è z sin exp i è z 0 0 cos exp i z è sin exp i z è cos exp i è z sin exp i è z cos exp i z è sin exp i z è cos sin exp i sin cos exp i sin cos, ès y è cos exp i è z sin exp i è z 0 i i 0 cos exp i z è sin exp i z è cos exp i è z sin exp i è z icos exp i z è isin exp i z è icos sin exp i isin cos exp i sin sin, ès z è cos exp i è z sin exp i è z 0 0 cos exp i z è sin exp i z è cos exp i è z sin exp i è z cos exp i z è sin exp i z è cos sin cos. Hence we see that S â ès x è x ès y è y ès z è z, as a vector in coordinate space, coincides exactly with the Bloch vector. It is important to remember that orthogonal states in H are represented by antipodal points on the Bloch sphere that is, points with angular separation. In Hilbert space or in coordinate space, of course, orthogonal vectors have angular separation /. Hence the division by in equation (6), which defines,. The origin of this discrepancy has to do with the fact that the operators S u correspond to a spinor representation of the 3D rotation group on a D complex vector space (more on this next term). 3
4 Dynamics on the Bloch Sphere static fields As long as we are concerned only with the spin degree of freedom for a spin particle (ignoring particle motion), Hamiltonian dynamics depend only on the particle s gyromagnetic ratio and the applied magnetic field B : H S B S x B x S y B y S z B z. When B is static (timeindependent), we are free to choose a coordinate system in which the z axis corresponds to the direction of B. Then H S z B z L S z, 3 where L B z is known as the Larmor frequency, and the energy eigenstates simply correspond to the S z eigenstates, H z è z è, L, H z è z è, L. 4 The time evolution of an arbitrary initial state t 0 è cos 0 exp i 0 z è sin 0 exp i 0 is thus given by t è cos 0 exp i 0 exp i t/ z è sin 0 exp i 0 exp i t/ z è, cos 0 exp i t z è sin 0 exp i t z è, 5 z è, 6 where t 0 L t. 7 Hence, we find that the Bloch vector (from the center of the Bloch sphere to the point representing t è) simply precesses around the z axis with angular frequency L. To see this another way, we can write S â ès x è x ès y è y ès z è z sin 0 cos t x sin 0 sin t y cos 0 z, 8 which we may also use to denote that the instantaneous direction of the spin s precession corresponds to that of S B (note that is negative for a bare electron, but may be either positive or negative for a composite spin particle such as an atom or nucleus). In fact, we may write d S S B, 9 in perfect agreement with the classical equations of motion for a magnetic moment in a static magnetic field! Since this basic dynamical picture should be independent of the coordinate system we 4
5 have chosen, we may conclude in general that the evolution of a spin particle in an applied magnetic field B corresponds to Larmor precession of the spin around B with angular frequency L B. Moreover, this precession should proceed precisely as predicted by the vector equation (9). Hence, if 0 è z è and B x,b y,b z B 0,0,0, we can guess that (assuming 0) t è sin Lt/ z è icos Lt/ z è. 0 Here we have inferred the relative phase of exp i i from the geometric picture that t èshould proceed from z è to z è through y è. We may compute explicitly, 0 è z è x è x è, t è exp i Lt/ x è exp i Lt/ x è exp i Lt/ z è z è exp i Lt/ z è x è isin Lt/ z è cos Lt/ z è i sin Lt/ z è icos Lt/ z è, which agrees with our prediction (0), up to overall phase. The basic phenomenon of Larmor precession is quite useful in experiments where one needs to detect, e.g., the presence of spin particles in a given volume of space. One standard detection method is to apply a large magnetic field B perpendicular to the expected direction of S. The spins will then precess at an appropriate Larmor frequency. Since the spins project a magnetic field pattern whose orientation is determined by S, Larmor precession also implies a period modulation in the magnetic field flux through any plane containing B. Using an inductive pickup coil, this periodic flux modulation can be detected via the induced EMF. According to Lenz s Law, this induced EMF should increase with the rate of change of the magnetic flux through the pickup coil, which in turn should increase with L. Hence, large B translates into high sensitivity in such detection methods. Note that in both of the Larmor precession examples above, t èis only periodic in L up to an overall minus sign. That is, when t / L, t è 0 è. This sign flip, which is sometimes known as the spinor property of spin state vectors, is not just some artifact of our definitions of, on the Bloch Sphere it is real, and can be observed in experiments. To see how this is possible, consider a composite system involving both a spin particle and an auxilliary twodimensional quantum system. We ll denote the spin Hilbert space by H A and the auxilliary Hilbert space by H B. What we need is to arrange a situation where the overall Hamiltonian is given by H AB B èè B å S z B z, where 0 B è, B è is an orthonormal basis for H B and S z acts on H A only. This type of Hamiltonian could be realized, for example, if H B corresponds to something like the position of the spin particle being inside or outside of a region of applied magnetic field B z. Then if the initial state 5
6 AB 0 è 0 B è B è å x è 0 B è å x è B è å x è 3 is prepared, the state at a time t / L later should be AB t è 0 B è å x è B è å x è 0 B è B è å x è. 4 Hence è AB t AB 0 è 0, 5 purely by virtue of the spinor property of the spin degree of freedom! Before moving on to timedependent magnetic fields, let us note the following fact. If we start out with a static applied field B B 0 z 6 and a spin particle prepared in the state z è (or z è), there is no finite perturbation W S B (where B lies in the x y plane) we can add to H 0 S z B 0 such that the particle will eventually evolve into the state z è (or z è). Simply put, this is because for nonzero B 0 there is no finite B such that B tot B 0 z B is orthogonal to the z coordinate axis. Hence d S S B tot 7 will never map z è into z è (or viceversa). Since one commonly utilizes large holding fields B 0 in order to achieve strong induction signals in spindetection experiments (as described above), it would appear that more sophisticated control measures must be applied in order to do things like flip the spins in an experimental sample. This leads us now to timedependent magnetic fields and the phenomenon of magnetic resonance. Timedependent magnetic fields In the previous section we examined the phenomenon of Larmor precession, which for a static applied magnetic field B B 0 z leads to state evolutions of the form t è cos 0 t exp i z è sin 0 t exp i z è 8 with t 0 Lt 0 B 0 t. Formally, this state evolution corresponds to Hamiltonian evolution that can also be described as the effect of a unitary timedevelopment operator: t è exp ih 0 t/ 0 è exp i L z t/ 0 è. 9 In what follows we shall assume the existence of a fixed holding field B B 0 z at all times. 6
7 Accordingly, it will be convenient to work in a rotating frame that formally eliminates the constant Larmor precession. Although it may not be obvious that this is worth the trouble, it is. Geometrically, we can think of defining timedependent coordinate axes x t x cos t y sin t, y t x sin t y cos t, z t z, 30 such that if we set L we may expect S x t sin cos t cos L t sin sin t sin L t, S y t sin cos t sin L t sin sin t cos L t, S z t cos, 3 all to be constant. Indeed, since cos t cos 0 L t cos 0 cos L t sin 0 sin L t, sin t sin 0 L t sin 0 cos L t cos 0 sin L t, 3 we have S x t sin cos 0 cos L t sin 0 sin Lt cos Lt sin sin 0 cos Lt cos 0 sin Lt sin Lt sin cos 0, S y t sin cos 0 cos L t sin 0 sin L t sin L t sin sin 0 cos L t cos 0 sin L t cos L t sin sin In terms of the quantum state vector, however, it would appear that we should define t è exp i L z t/ t è â O z t è 34 (here we are defining the unitary operator O z ). Then t è 0 èas long as the Hamiltonian is given simply by H 0 L S z. 35 However, in order to continue using t èin the presence of perturbations, we need to derive its general equation of motion in the rotating frame. Let us also return to a general setting in which the frequency that defines the rotating frame via t è O z t è exp i z t/ t è 36 is independent of L. We can start from the Schrödinger Equation i d t è H t è HO z t è. 37 7
8 On the lefthand side, we may also apply the chain rule to yield i d t i d O è z t è i do z t è O d z t è. 38 Hence, combining the two expressions and multiplying through from the left by O z,we obtain i d t è O z HO do z io z z t è â H t è. 39 Recall that since O z is unitary, O z Oz. Hence do io z z i exp i d z t/ exp i z t/ i exp i d z t/ n! i z t/ n i exp i z t/ n i exp i z t/ n n0 n n! i z t/ n i z / n! i z t/ n i z / i exp i z t/ exp i z t/ i z / z. 40 Now we can finally make use of all this to solve for the effects of a timedependent perturbation, W t b x cos t y sin t S b cos t x sin t y. 4 Here b is a magnetic field strength. At this point it is convenient to define new operators x i y, x i y, 4 in terms of which W t b exp i t exp i t. 43 Noting the commutation relations 8
9 , z x, z i y, z i y x,, z x, z i y, z i y x we have H, 44 O z HO do z io z z O z H 0 W O z z O z L z b exp i t exp i t O z z ' z b exp i z t/ exp i t exp i t exp i z t/, 45 where ' â L and we have used the fact that z,o z z,exp i z t/ 0. Next we need to compute exp i z t/ n0 n0 n0 n! i z t/ n n! i t/ n z n n! i t/ n n exp i t/, 46 where in going from the second to the third line we have used the fact that z x i y z Similarly, exp i z t/ exp i t/, exp i z t/ exp i t/, exp i z t/ exp i t/, 48 9
10 and we finally arrive at H ' z b ' z b x where S B eff, 49 B eff B 0 z b x. 50 Recall that L B 0, so if the z component of B eff vanishes. And in general, we have the simple result that if ' â L, the magnitude of the effective holding field is greatly reduced in the rotating frame. In particular, if '0then the Bloch vector corresponding to t èshould simply precess around a static field b x in the rotating frame. It thus follows that for an initial state such as 0 è z è, the application of a rotating field B t b x cos L t y sin L t 5 will lead to t è z è at time t / b. This technique enables perfect spin flips even in the presence of a large holding field B 0 z. Note however that if ' â 0, it is still impossible to achieve z è z è with finite b. Hence it is crucial to apply the rotating field B t exactly at the resonance frequency L. As long as this condition is met, z è z è will happen eventually, even for very small b (as long as we ignore dissipation!). In this sense we find that the frequency of the applied perturbation is much more important than its magnitude, at least for the purpose of perfectly flipping spins. This is generally referred to as the phenomenon of magnetic (or twolevel) resonance. Dynamics in the rotating frame Having derived the form of the effective Hamiltonian H ' z b x 5 in the rotating frame, we are able to apply our earlier results about the general behavior of twolevel systems under static perturbations. We may define as well as H 0 H 0 ' z, W b x, 53 E è, E ', H 0 E è, E ', 54 where è Û z è, è Û z è. 55 Then, e.g., the eigenvalues of H are given by 0
11 E E W E W E W E W 4 W. 56 With these exact expressions for the energy eigenvalues and eigenvectors, it is straightforward to compute the timeevolution of an arbitrary initial state in the rotating frame. We just have to be careful to transform back to the static (laboratory) frame at the end of the calculation, via t è O z t è. 57
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