Saturation of a spin 1/2 particle by generalized Local control

We show how to apply a generalization of Local control design to the problem of saturation of a spin 1/2 particle by magnetic fields in Nuclear Magnetic Resonance. The generalization of local or Lyapunov control arises from the fact that the derivative of the Lyapunov function does not depend explicitly on the control field. The second derivative is used to determine the local control field. We compare the efficiency of this approach with respect to the time-optimal solution which has been recently derived using geometric methods.


Introduction
The control of spin systems by magnetic fields has a great potential for applications in Nuclear Magnetic Resonance (NMR) [1]. An example is given by the preparation of a well-defined initial state such as the saturation process which consists in vanishing the magnetization vector of a sample by using an adequate pulse sequence [2,3]. In the setting of NMR spectroscopy and imaging, the saturation control can be used for solvent suppression or contrast enhancement. Different approaches have been proposed up to date to solve this control problem. They extend from intuitive schemes such as the Inversion Recovery (IR) method [1] to more elaborate pulse sequences based on geometric optimal control theory [4] or optimization schemes [5,6,14,17]. In a recent letter [4], we showed that a gain of 60 % in the duration of the saturation process can be reached when using the time-optimal solution with respect to the IR one. In general, however, analytic solutions as derived in [4] are not available, and easily accessible numerical methods are required for the control of complex and realistic systems. Here, we investigate such a method and demonstrate its efficiency through a comparison with time-optimal analytic solutions [4]. Roughly speaking, the present method is an extension of local [7] or Lyapunov control [8], in terms of a so-called Lyapunov function V over the state space which is minimum (or maximum) for the target state. In original Lyapunov control theory, a control field that ensures the monotonic decrease of V is constructed with the help of the first time derivativeV of V . Under well established general conditions on the Hamiltonian (see e.g. [9] for a recent review), the system converges asymptotically towards the target state. This approach has been largely explored in quantum mechanics both from the physical [7,11,10] and mathematical points of view [12].
In this paper, we will employ a target functional whose first time derivative does not depend on the available control fields. In this case, an inspection ofV does not help to identify suitable control fields and standard local control theory cannot be used. Therefore, one has to resort to the second time derivativeV , which allows the design of control pulses that speed up the increase of V [13]. On the first sight, one might assume that loosing control ofV should be a disadvantage. However, it turns out that it also yields significant advantages: since control fields typically induce unitary dynamics, the independence ofV on the control field implies the invariance of V under the corresponding unitary group. Consequently, the function V is constant for a continuous manifold of quantum states within which the control can induce dynamics. By not enforcing a given path through state space, but leaving the systems the opportunity to find its optimal path with a direction restricted only by the manifolds of constant V permits to not exert unnecessary control, but to leave the system substantial freedom to find its optimal path. Also,V allows to predict the dynamics of V for longer intervals thanV , since it extrapolates for two infinitesimal time steps rather than one. This reduces the time-local character of Lyapunov control, and the efficiency of this generalized approach will be illustrated by different numerical simulations in this work.
The paper is organized as follows. Section 2 deals with the description of the system and of the control problem. The description of the generalized local control approach and of the time-optimal one is done in Sec. 3. Different numerical results are finally presented in Sec. 4 for relevant situations in NMR. Conclusion and prospective views are given in Sec. 5.

The model system
The state of a spin 1/2 particle in an NMR experiment is most conveniently expressed by its Bloch vector with components M x , M y and M z . If the radiofrequency magnetic field is in resonance with the spin, the Bloch equations are given by: where ω x and ω y are the two components of the control field ω, T 1 and T 2 are two relaxation rates characterizing the incoherent process due to the interaction with the environment and M 0 is the thermal equilibrium magnetization. Since the maximum amplitude ω max that the control fields ω can adopt induces a natural time-scale, it is convenient to introduce the scaled variables Since, in addition, the equations of motions are invariant under rotations around the z-axis, we can restrict the entire dynamics to the (y, z)-plane by assuming that ω y = 0. Doing so, Eqs. (1) reduce tȯ in terms of the scaled and scalar control field u = 2πω x /ω max .

Optimization schemes
In the following we will take the thermal equilibrium state with M z = M 0 and M x = M y = 0 as initial state. In terms of the scaled magnetization, this implies that the initial state is located at the north pole of the scaled Bloch sphere. As mentioned below, the goal of the control is to bring the scaled Bloch vector to the center of the sphere in minimum time with the constraint |u| ≤ 2π on the control field.

Generalized local control theory
Since our goal is to reach a vanishing magnetization vector, which is equivalent to a maximum entropy state, we can choose any entropy-like functional as target.
We consider the linear entropy defined as Reaching a maximum entropy in minimum time, i.e. as quickly as possible, can be done by choosing u such that the temporal increment of S l is maximized. As anticipated above, the time derivative of S l is independent of u, which is a direct consequence of the invariance of S l under coherent dynamics, as generated by u. We, therefore, have to inspect the curvature of S l , which essentially contains the information of how the impact of relaxation changes under the coherent dynamics. In the present case, it reads whereS o l denotes the curvature in the absence of a control field. Since maximization of the curvature results in maximally fast increase ofṠ l , which in turn implies an acceleration of the increase of entropy, we will choose, at any instance of time, u such that it maximizes the curvature. Given that the modulus of u is bounded by 2π, this leads to the choice u = 2π if µ > 0, and u = −2π if µ < 0. As long as µ is finite, this implies that u is constant over a finite period of time so that the equations of motion can be integrated straightforwardly. Care is, however, necessary if µ = 0. In this case, one has to distinguish between stable solutions, where the control fields ensure that the condition µ = 0 is preserved and unstable conditions, where the control, or the natural dynamics ensures that the condition µ = 0 is satisfied only for an isolated point in time. The latter case is realized for y = 0 and it corresponds to u = 0 (i.e. no control pulse), the system remaining in the line of equation y = 0 to follow its natural dynamics. The former case is realized for z = z 0 = γ/(2(γ − Γ)) and requires some more care. Let us, therefore, step back for a moment from the framework of differential equations and discuss Eqs. (2) in terms of finite time steps. If we start out with µ ≤ 0 than the value of µ will increase in time since we were assuming the condition µ = 0 to be stable. Due to the finite time steps, however, we might end up in a situation where µ ≥ 0. Now, in turn, µ will decrease in time, so that we risk to end up in rapid changes of sign of µ, which results in a rapidly changing sign of the control pulses. Decreasing the size of the time steps will decrease the size of the fluctuations of µ and increase the frequency with which the control pulse changes sign. Once this frequency has exceeded all system frequencies sufficiently, we can replace the rapidly oscillating pulse by a time-averaged pulseū where f (t) is some function, like a Gaussian, that is positive in some finite domain around t = 0, and that decays sufficiently fast outside this domain. In the limit of infinitesimal time steps, the fluctuation of µ will vanish, andū is the pulse that has to be applied to preserve the condition µ = 0. The controlū needs to be chosen such thatż This implies that the local pulse solution can be written as ) .
Note thatū → ±∞ when y → 0 which defines a limit of admissibility due to the bound on the control given by This means that for smaller values of y, the system cannot follow this horizontal line and a constant control u = ±2π has to be used. Together with Eq. (2), the equation of motion for the y-component with u =ū reads consequently aṡ which has the solution where y 0 is the initial y-coordinate on the horizontal axis. One deduces thatū is determined through Eq. (8), and,ū in turn, defines z(t) uniquely. One finally arrives at: u = 0 for y = 0 and µ = 0, u =ū as given by Eqs. (8) for µ = 0 and z = z 0 , and the complete solution is a concatenation of these pieces, as sketched in Fig. 1. In the following, the control fields of maximum intensity will be called regular or bang, while those such that |u| < 2π will be said to be singular.  15); the dotted arrow corresponds to free evolution according to Eq. (14). The curved, solid arrows correspond to Eqs. (12) and (13).
Using this material, we can follow the dynamics starting from the thermal equilibrium (y = 0, z = 1). This point is an unstable fixed point of the dynamics, i.e. following the above prescription one should not add any control pulse, the state being stationary. Any small deviation, however, will indicate that a finite control field needs to be applied. We will, therefore, begin with an infinitesimally displaced initial condition y(0) = −ε, z(0) = 1. Then, according to Eq. (13) the system is driven with maximum intensity u = −2π until the Bloch vector reaches a point with µ = 0. At this point, the control field has to be changed such as to satisfy Eq. (15) and the Bloch vector moves along the line µ = 0 until its y-component vanishes. Once, this is achieved, the spin will relax to the target state without any control field according to Eq. (14). For the moment, we have been assuming that the control field can be chosen strong enough to ensure that the spin remains on the horizontal µ = 0-trajectory. Here, it is not the case due to the limitation in the field amplitude, a constant pulse according to Eq. (13) has therefore to be applied whenū = ±2π along the horizontal line.

Optimal control theory
In this section, we briefly recall the tools of optimal control theory based on the application of the Pontryagin Maximum Principle (PMP). The reader is referred to [16] for a general application to dissipative quantum systems and to [4,15,17] for examples in NMR. The first step of this method consists in introducing the pseudo-Hamiltonian H defined by where X = (y, z) and the vectors fields F 0 and F 1 have respectively the components (−Γy, γ(1 − z)) and (−z, y). The vector P is the adjoint state of coordinates (p y , p z ). The PMP states that the optimal trajectories are solutions of the system˙ Introducing the switching function Φ = P · F 1 , the PMP leads to two types of situations, the regular and the singular ones. In the regular case, Φ( X, P ) = 0 or vanishes in an isolated point while in the singular case Φ is zero on a time interval. In the former situation, the corresponding control field is given by u = 2π × sign [Φ]. In the latter case where φ =φ =φ = · · · = 0, we introduce the vector V = (−γ + γz − Γz, (−Γ + γ)y) which satisfiesφ = P · V . The singular trajectories belong to the set of points where V is parallel to F 1 , i.e. to the points such that det( F 1 , V ) = 0. A straightforward computation leads to the two lines of equation y = 0 and z = z 0 [4], which shows that the singular set is the same in the local control and optimal approaches. The corresponding optimal singular control is by definition the field such that the dynamics remains on the singular lines. Still this field is identical to local singular control. At this point, it is important to note that the trajectories given by the maximization of H are only extremal solutions, i.e. candidates to be optimal. Other tools like those introduced in [18] have to be used to get optimality results and to select among extremal trajectories the ones which are effectively optimal.

Comparison of local and optimal control sequences
As a specific example to illustrate the efficiency of these two optimization schemes, we consider the control problem of a spin 1/2 particle coupled to a thermal environment as analyzed and experimentally implemented in Ref. [4].
In this section, we will consider two examples characterized by different relaxation parameters. In the first case, we have T 2 = 8.8 ms and T 1 = 61.9 ms which leads, with a maximum amplitude ω max = 2π × 32.3 Hz, to Γ = 3.5 and γ = 0.5. In the second situation, we choose T 2 = 6.2 ms and T 1 = 61.9 ms which gives with the same maximum amplitude Γ = 5 and γ = 0.5. Let us compare the time-optimal solution as computed in [4,16] with the solutions obtained with the generalized local control theory. The numerical results are represented in Fig. 2 and 3 for the first and second cases, respectively. In the two cases, the time optimal solution is composed of two bang pulses of maximum amplitude 2π and two singular controls: starting from the initial point of coordinates (y = 0,z = 1), the first bang pulse drives the spin up to the horizontal singular line where µ vanishes. The consecutive singular control ensures that the z-component remains constant until y = y c , when the second bang pulse is applied. The position y c is computed to ensure that the switching function φ and its first derivativeφ remain continuous along the optimal sequence. The values of y c are equal to -0.1605 and -0.1356 in the first and second cases. The second bang pulse increases the z-component of the magnetization vector until its y-component vanishes. The final part of the optimal sequence is a zero control along the vertical singular axis y = 0 which allows to reach the target state. With the chosen relaxation parameters γ and Γ, the time-optimal sequence has a duration of 0.95050 and 0.97 in dimensionless units [defined above in Eq. (2)] in the first and second cases. When there is no bound on the control, the second bang does not exist and the optimal sequence is formed by a bang arc followed by the two singular trajectories along the horizontal and vertical lines. The solution of the generalized local control is very similar. Exactly like the time-optimal solution, it begins with a bang pulse followed by a singular pulse ensuring that µ vanishes, but this pulse is maintained longer than in the timeoptimal solution. Only at y = −0.0862 and y = −0.0840 where |u| = 2π, the limited control intensity does not permit to maintain the condition µ = 0 and a bang pulse has to be applied. In the unbounded situation, note that the local and optimal solutions are identical. In the bounded case, if the second bang arc intersects the vertical z-axis, the dynamics relaxes without any control until its Bloch vector vanishes. In the case of Fig. 2, this leads to an overall control duration of 0.95098 which is slightly larger than the one of the time-optimal solution, the difference is of the order of 4.81 × 10 −4 . Whereas local control does not reproduce the time-optimal solution exactly, we can conclude in this case that the longer duration of the former is negligible for all practical purposes. This conclusion is not true for all the values of the relaxation parameters. In particular in Fig. 3, since there is no intersection between the second bang pulse and the z-axis, we observe that the local control approach does not manage to reach the target, the minimum distance being equal to 9.9 × 10 −4 . Figure 4 gives a third example where the initial point of the dynamics is the south pole of the Bloch sphere, the relaxation rates being the ones of the first example. If we assume that the initial point is exactly on the z-axis then the local solution consists of letting act the relaxation with u = 0 up to the center of the Bloch ball. Since this axis is unstable, any small deviation from this point will induces a bang control of amplitude ±2π up to the horizontal singular line. A second bang pulse is applied when the limit of admissibility of the control is reached and a zero control finally used up to the target state. The optimal control sequence is quite similar, except for the second bang which is applied earlier than in the local control strategy. We observe in Fig. 4 that the local and the optimal solutions are very close to each other with a control duration respectively of 0.8758 and 0.8753, while if u = 0 this time is of 1.3861.

Conclusion and perspectives
The close-to-optimal performance of the generalized time-local control pulses reflect the potential of this new approach which can efficiently replace the standard local control design when this strategy cannot be used. Whereas we have seen that sufficiently strong control typically permits to reach the target, it can also be missed with weak control fields. This opens up the general question of convergence in the present generalized framework of Lyapunov control. In standard Lyapunov control theory there is a well-established theory of convergence, based, e.g., on the concept of the Lasalle invariant set on which the derivative of the Lyapunov function vanishes [19]. In addition, necessary and sufficient conditions on the Hamiltonian of the system have been derived to ensure the converge towards the target state [9]. Given the success of these tools for standard Lyapunov control theory it seems within reach to also develop rigorous concepts to predict convergence properties of the present extension.