Bloch sphere

Figure Al.6.32. (a) Initial and (b) final population distributions corresponding to cooling, (c) Geometrical interpretation of cooling. The density matrix is represented as a point on generalized Bloch sphere of radius R...

The basic element of a quantum computer is the quantum bit or qubit. It is the QC counterpart of the Boolean bit, a classical physical system with two well-defined states. A material realization of a qubit is a quantum two-level system, with energy eigenstates, 0) and 1), and an energy gap AE, which can be in any arbitrary superposition cp) = cos(d/2) 0) + exp(i0)sin(0/2) l).These pure superposition states can be visualized by using a Bloch sphere representation (see Figure 7.1). 

Fig. 6. Simulation of the effect of longitudinal and transverse relaxation during a 30 ms selective inversion pulse. The trajectories are shown on a Bloch sphere and in the...

The salient features of the dynamics of our model molecule are best exhibited with the help of Poincare sections that have already proved useful in the analysis of the double pendulum presented in Section 3.2. Fig. 4.7 shows the rpp projection of an x = 0 surface of section of a trajectory for a = 0.1, uq = 10 and E = 4 started at 0 = 0.957T, x = sin(0), y = 0, z = cos(0) and tj = 1.42. The resulting y-p Poincare section clearly shows chaotic features. This indicates that the classical dynamics of the skeleton of the model molecule is chaotic. But the most striking feature of the model molecule is its fully chaotic quantum dynamics. This is proved by Fig. 4.8, which shows the chaotic quantum fiow of the molecule on the southern hemisphere of the Bloch sphere. Fig. 4.8 was produced in the following way. First we defined the Poincar6 section by p = 0, dp/dt > 0. Then, we ran 40 trajectories in x,y,z,r],p) space for a = 0.1, Uo = 10 and E = Q starting at the 40 different initial conditions... 

Xj = sin(0, ), Vj = 0, Zj = cos 9j), rjj = 2.4, 9j = y7r/40, j = 1,2,..., 40. In order to represent the resulting section points in the plane, we projected them onto the x — y plane (the equatorial plane) of the Bloch sphere. In order to obtain a unique representation we divided the Bloch sphere into a northern and a southern hemisphere according to z > 0 and z < 0, respectively. Only the projections of points with z <0 ( southern hemisphere ) are shown in Fig. 4.8. We see that the southern hemisphere of the Bloch sphere is mostly chaotic. But this means that in the chaotic sea of Fig. 4.8 the quantum d30iamics of the model molecule is genuinely chaotic. 

In order to prove that the quantum dynamics on the Bloch sphere of the model molecule is genuinely chaotic, Bliimel and Esser (1994) calculated the Euclidean distance d(r) = [(a (r) — x t)) -b (y(r) — y r)) + (z(r) — /(r)) ] / between two initially close trajectories. The result is shown in Fig. 4.9. It proves that for r < 300 the distance d(r) grows exponentially. This corresponds to a positive Lyapunov exponent for the quantum subsystem. At r 300 the exponential growth of d(r) breaks. 

Fig. 4.8. Quantum chaos on the southern hemisphere of the Bloch sphere of the model molecule. (Prom Bliimel and Esser (1994).)...

This is natural since d cannot be larger than the diameter of the Bloch sphere, which equals 2. 

Hennig, D. and Esser, B. (1992). TYansfer dynamics of a quasiparticle in a nonlinear dimer coupled to an intersite vibration Chaos on the Bloch sphere, Phys. Rev. A46, 4569-4576. 

Recall again, that it is quite tricky to use two-level systems (instead of, say, an ammonia molecule) in individual quantum theory. Nevertheless, two-level systems can be quite instructive, precisely because simple visualization is possible by means of the Bloch sphere. 

In Fig. 5 some of these different decompositions are visualized with the aid of the Bloch sphere. The thermal density operator Dp= from... 

FIGURE 5 A thermal density operator can be decomposed into pure states in infinitely many different ways. Mbdng the vectors corresponding to eigenstates or chiral states or alternative chiral states with 50% probability always leads to the zero vector, i.e., the center of the Bloch sphere, corresponding to the density operator Dp =... 

Eq. (26) corresponds to the zero vector in three-dimensional space, i.e., to the center of the Bloch sphere. Statements about the decompositions of Dp made earlier can be verified by computing the vectors b., b, b, b, i>, and b which correspond to the respective pure state vectors 1, and This can be done by using Eq. (25). The chiral states,... 

It is, of course, not compulsory to decompose a thermal density operator into only two or finitely many pure states. One could equally well try to decompose a thermal density operator into a continuum of pure states, or even into all the pure states of the system in question. This possibility is illustrated here by use of a two-level system and the relevant Bloch sphere. 

FIGURE 7 A subset B of all the pure states of a two-level system can be specified by giving the corresponding subset B of the surface Sj of the Bloch sphere. 

During this tunneling process, the nuclear molecular framework is not conserved in between the alternative chiral states (1/ /2 )[ h arise, which do not possess a nuclear structure. Incidentally, for small level splitting (E -E ), the tunneling process is very slow and so we need to ask which of the available chiral states (on the equator of the Bloch sphere) actually arise in a properly chiral molecule. 

The pure state of the molecular environment is never precisely known. Since the dynamics of the molecular pure state depends on this unknown environment s (pure) state, one gets a stochastic dynamics for the molecular pure states. For a two-level system-with its pure states describable by a Bloch sphere—the situation is illustrated in Fig. 8. The dynamics of some given molecular initial state is governed not only by the Schrodinger equation, but also by external influence. Depending on the (pure) state of the environment, we reach different final molecular states. Usually only probabilistic predictions can be given (and no information about the precise trajectory of pure states, i.e., the trajectory on the Bloch sphere). 

Pig, 4. An RF pulse initially rotates the magnetization away from the z-axis (i.e. thermal equilibrium) by, in this example, an angle of about 170°. In the absence of radiation damping effects the magnetization follows the normal relaxation pathway (—) back to the thermal equilibrium. However, in the presence of radiation damping, the magnetization retains its coherence and returns to the equilibrium position on the surface of the Bloch sphere (---). (After Mao et al )... 

This way of writing A and B seems arbitrary, but it is not. It is the same parametrization that is used in constructing the Bloch sphere, which is a powerful tool in the analysis... 

However, it should be noted that there is no general solution for the problem of the movement of the damped Bloch vector (Allen and Eberly 1975). So in general, the Bloch sphere is considered only for time t MF. 

This representation allows a geometric visualization of the qubit quantum state as a point on the surface of a unit radius sphere, called Bloch sphere. The most important points on Bloch sphere are shown on the table below, adapted from Ref. [1]. 

An appropriate and useful approach to follow the evolution of a quantum state is the Bloch sphere representation, introduced in Chapter 3. This is a geometrical scheme in which the quantum state and its evolution is represented by the trajectory of a vector over the so-called Bloch sphere (Figure 4.13). In the Bloch sphere, the poles represent the two eigenstates of the system, whereas the equatorial plane corresponds to an uniform superposition of these two eigenstates. 

Figure 4.13 Schematic representation of a state vector in the Bloch sphere.

To illustrate the Bloch sphere representation of an NMR system, let us consider the density matrix of an ensemble of spins 1/2 nuclei. Because the high temperature deviation density matrix is proportional to 1, the effect of an RF pulse is to induce rotations that transform the initial density matrix into a linear combination of the spin operator components,... 

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