Time-minimal control of dissipative two-level quantum systems: The integrable case
The objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. We focus our analysis on the case where the extremal Hamiltonian is integrable.
Keywords. Optimal control, conjugate and cut loci,
AMS classification. 49K15, 70Q05
We consider a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation which takes the following form in suitable coordinates , i.e., in the coherence vector formulation of density matrix [1, 16]:
We refer to  and  for the details of the model. We recall that and are related to off-diagonal terms of the density matrix of the system and to the difference of population between the two states. In (1), is a set of parameters such that and which describes the interaction of the two-level system with the environment. More precisely, is the dephasing rate and and are respectively equal to and where the coefficients and are the population relaxation rates. The control is the complex Rabi frequency of the laser field which is assumed to be in resonance with the frequency of the two-level system . The physical state belongs to the Bloch ball, , which is invariant for the dynamics considered. If they are many articles devoted to optimal control of quantum systems in the conservative case (see e.g. ), the dissipative case is still an open problem.
The system can be written shortly as a bilinear system
and in order to minimize the effect of dissipation, we consider the time minimum control problem for which, up to a rescaling on the set of parameters , the control bound is . The energy minimization problem with the cost where the time is fixed but the control bound is relaxed can also be considered and shares similar properties.
A first step in the analysis of such systems is contained in . Assuming real, the problem can be reduced to the time-optimal control of a two-dimensional system
with the constraint . For such a problem, the geometric optimal control techniques for single-input two-dimensional systems presented in  succeed to make the time-optimal synthesis for every values of parameters .
In order to complete the analysis in the bi-input case, a different methodology has to be applied and we shall make an intensive use of techniques and results developed in a parallel research project to minimize the transfer of a satellite between two elliptic orbits (see ). Such techniques are two-fold.
First of all, the maximum principle will select extremal trajectories, candidates as minimizers and solutions of an Hamiltonian equation. A geometric analysis will identify the symmetry group of the system and find suitable coordinates to represent the Hamiltonian. A consequence of this analysis is the fact that, in the case , the extremal system is integrable and if , the problem can be in addition reduced to a 2D-almost Riemannian problem on a two-sphere of revolution for which a complete analysis comes from . We take advantage of this property to analyze the general integrable case using continuation methods on the set of parameters, while the analysis fits in the geometrical framework of Zermelo navigation problems . This case corresponds to the physical situation where the population relaxation rates are equal. The mathematical tools presented in this article and a few numerical simulations are sufficient to complete the analysis for . The generic case where is treated in a forthcoming article  combining mathematical analysis and intensive numerical computations.
Secondly, having selected extremal trajectories, second-order conditions using the variational equation and implemented in the cotcot code  allow to determine first conjugate points forming the conjugate locus which are points where extremals cease to be locally optimal. Combined with the geometric analysis, we can construct the cut locus which is formed by points where extremals cease to be globally optimal.
The organization of this article is the following. In section 2, we recall the maximum principle and the concept of conjugate points associated to second-order optimality conditions. In section 3, we present the geometric analysis of the system, followed in section 4 by a thorough analysis of the so-called Grusin problem on a two-sphere of revolution. This problem is generalized into a Zermelo navigation problem suitable to our analysis. In section 5, we study the properties of the extremals to analyze the optimal trajectories, combining analytic and numerical methods.
2 Geometric optimal control
2.1 Maximum principle
We consider the time minimum problem with fixed extremities and . For a smooth system written , with and a control domain which is a compact subset of , we have:
If is an optimal control trajectory pair on then there exists an absolutely continuous non-zero vector function such that almost everywhere on , we have:
where , being a non positive constant; is defined by and is zero everywhere.
Application: We consider the time-minimum control problem for a system of the form with , , . We introduce the Hamiltonian lifts , of the vector fields and the set such that for . Then the maximization condition (5) leads to the following result.
Outside , an extremal control is given by , and extremal pairs are solutions of the smooth true Hamiltonian vector field with .
The surface is called the switching surface and the solutions of are called extremals of order zero. To be optimal, they have to satisfy and those with are called abnormal.
An important but straightforward result is the following proposition .
Extremal trajectories of order zero correspond to singularities of the end-point mapping
where denotes the response to with initial condition and such that the control is restricted to the -sphere .
2.2 Second-order optimality conditions
From proposition 3, we can apply the concepts and
algorithms presented in  to compute second-order
optimality conditions in the smooth case, when the control domain
is a manifold, being restricted to the unit sphere. The
of this computation is next recalled.
The concept of conjugate point:
Since is a manifold, we may assume locally that and the maximization condition (5) leads to
Our first assumption is the strong Legendre-Clebsch condition:
(H1) The Hessian is negative definite along the reference extremal.
From the implicit function theorem, an extremal control can be locally defined as a smooth function of and plugging into defines a smooth true Hamiltonian .
Setting and using Hamiltonian formalism, we introduce:
Let be a reference extremal defined on . The variational equation
is called the Jacobi equation. A Jacobi field is a non trivial solution . It is said to be vertical at time if .
The following standard result is crucial.
Let be the fiber and let be its image by the one parameter subgroup generated by . Then is a Lagrangian manifold whose tangent space at is spanned by Jacobi fields vertical at . Moreover, the rank of the restriction to of the projection is at most .
We next formulate the relevant generic assumptions using
the end-point mapping.
(H2) On each subinterval , , the singularity of is of codimension one for .
(H3) We are in the normal case .
As a result, on each subinterval there exists up to a positive scalar an unique adjoint vector such that is extremal.
We fix and we define the exponential mapping:
where is a dimensional vector, normalized with .
Let be the reference extremal on . Under our assumptions, a time is called conjugate if the mapping is not an immersion at and the point is said to be conjugate to . We denote the first conjugate time and the conjugate locus formed by the set of first conjugate points considering all extremal curves.
Let be a reference extremal on satisfying assumptions (H1), (H2) and (H3). Then the extremal is optimal in the -norm topology on the set of controls up to the first conjugate time . Moreover, if is one-to-one then it can be embedded into a set , image by the exponential mapping of , where is a conical neighborhood of . For , the reference extremal trajectory is time minimal with respect to all trajectories contained in .
In order to get global optimality results, it is necessary to glue together such micro-local sets. We need to introduce the following concepts.
Given an extremal trajectory, the first point where it ceases to be optimal is called the cut point and taking all the extremals starting from , they will form the cut locus . The separating line is formed by the set of points where two minimizers initiating from intersect.
3 Geometric analysis of Lindblad equation
3.1 Symmetry of revolution
If we apply to system (1) a change of coordinates defined by a rotation of angle around the axis:
and a similar feedback transformation on the control:
we obtain the system
Hence, this defines a one dimensional symmetry group and by construction . Therefore, we deduce that the time-minimum control problem (or the energy minimization problem) are invariant for such an action. Using cylindric coordinates
and the dual variables , the Hamiltonian takes the form
In particular, the Bloch ball is foliated by meridian planes in which the time-minimum synthesis is the one associated to system (3), where the control is scalar and described in . More precisely, we have:
For the time-minimum control, is a cyclic coordinate and is a first integral of the motion. The sign of is given by and if then is constant and the extremal synthesis for an initial point on the z-axis is up to a rotation given by the synthesis in the plane . Up to a rotation, the control can also be restricted to the single-input control .
The proof is a generalization of the geometric situation encountered in . For the Hamiltonian vector field , the points on the -axis correspond to a polar singularity and the extremals starting from the -axis are contained in meridian planes . Hence, is constant and extremal curves in the plane are solutions of system (3). ∎
3.2 Spherical coordinates
More properties can be seen using spherical coordinates:
and a similar feedback transformation. We obtain the system:
and the corresponding Hamiltonian
In this representation, are the spherical coordinates on the unit sphere of revolution around the z-axis: is the angle of revolution and is the angle of the meridian, correspond respectively to the north and south poles.
3.3 Lie brackets computations
In order to complete the analysis, we immediately compute the Lie brackets up to length 3 for the system written in Cartesian coordinates as:
where the ’s are the matrices
and is the vector . It can be lifted into a right-invariant control system on the semi-direct product identified to the subgroup of matrices of of the form:
acting on the subset of vectors of : To construct affine vector fields, we use the induced action of the Lie algebra and Lie brackets are given by
The control distribution is and we have:
In particular, we obtain that and hence the system on :
is controllable. For the linear action, it defines a controllable system on the unit sphere. This action has however singularities:
at 0, the orbit is 0.
the set on where and are collinear is the whole plane and restricted to the unit sphere of revolution, it corresponds to the equator.
To analyze the effect of the drift term associated to dissipation, we use
and . Moreover, we have
Those computations reveal the singularity at , that we describe in the next proposition.
In the case , , the radial component is not controllable and the time-minimum control problem is an almost Riemannian problem on the two-sphere of revolution for the metric in spherical coordinates with Hamiltonian .
If and then the first equation of (3.2) becomes and is not controllable. Hence, the time-minimal control problem reduces to the problem of controlling in minimum time where the associated true Hamiltonian is . The time minimum problem is equivalent to minimizing the length for the metric . According to Maupertuis principle, we can replace the length by the energy with corresponding Hamiltonian . ∎
The almost Riemannian metric is called the Grusin model on the two-sphere of revolution.
Such a metric appears in quantum control in the conservative case  and a similar metric is associated to orbital transfer . It will be analyzed in details in section 4 since it is the starting point of the analysis in the general case using a continuation method on the set of parameters.
Another consequence of the previous computations is the controllability properties of the system and the structure of extremal trajectories.
3.4 Controllability properties
We recall that the Bloch ball is invariant. Indeed, introducing , we get:
which is strictly negative on the unit sphere except if , and . Using the representation (3.2) of the system in spherical coordinates, it is clear that we can control the angular variables and if the controls are not uniformly bounded. If then we have restrictions depending upon the set of parameters.
For , the system is homogeneous and is a fixed point. The accessibility set in fixed time is with non empty interior for a non-zero initial point, except in the case which corresponds to the Grusin model and for which the time and energy minimization problem are equivalent.
The controllability properties for are clear in this case. Indeed, if then we can compensate the drift by feedback for the system on the two-sphere of revolution, while it is not the case for .
Let and be two points in the Bloch ball such that is accessible to . Then there exists a time-minimum trajectory joining to . Moreover, every optimal trajectory is
either an extremal trajectory with , contained in a meridian plane, time-optimal solution of the two-dimensional system (3) where .
either connection of smooth extremal arcs of order 0, solutions of the Hamiltonian vector field with , while the only possible connections are located in the equatorial plane .
The control domain is convex and the Bloch ball is compact. Hence, we can apply the Filippov existence theorem . In order to get a regularity result about optimal trajectories, much more work has to be done. This is due to the existence of a switching surface : in which we can connect two extremals arcs of order 0, provided we respect the Erdmann-Weierstrass conditions at the junction, i.e., the adjoint vector remains continuous and the Hamiltonian is constant. The set can also contain singular arcs for which holds identically. Hence, we can have intricate behaviors for such systems. In our case, the situation is simplified by the symmetry of revolution.
Indeed, if then the singularities are related to the classification of extremals in the single-input case, which is described in . We cannot connect an extremal with where is the global first integral to an extremal where since the adjoint vector has to be continuous.
Hence, the only remaining possibility is to connect two extremals of order 0 with at a point of leading to the conditions and in spherical coordinates. Since , one gets . The result is proved. ∎
Remark: The classification of extremal trajectories near the equatorial plane is described in proposition 19.
4 The Grusin model on a two-sphere of revolution with generalizations to Zermelo navigation problem
The Grusin model is a special case of metrics of the form on a two-sphere of revolution such that:
(H1) on .
(H2) (reflective symmetry with respect to the equator).
They appear in optimal control in the orbit transfer, smooth at the equator or with a polar singularity, in quantum control and in various geometric problems, e.g., Riemannian problems on an ellipsoid of revolution. The importance of this control problem has justified the recent analysis of  that we complete next, using Hamiltonian formalism in order to make generalizations. We first interpret the Grusin model as a deformation of the round sphere.
The standard homotopy between the Grusin model and the round sphere on the two-sphere of revolution is where , and .
By construction, the metric is analytic for and
for , we have the Grusin model with a pole of order 1
at the equator.
The first objective of this section is to show stability results concerning such metrics. We have the following general result .
Let be a smooth metric on a two-surface of revolution. Then,
Extremals are solutions of the Hamiltonian and arc-length parametrization amounts to restrict to .
If is the angle of an unit-speed extremal with a parallel then is a constant and the extremal flow is Liouville integrable with two commuting first integrals and .
The Gauss curvature is .
We next make a complete analysis of the family of metrics .
4.1 Curvature analysis
We have the following proposition.
For the family of metrics , we have:
The Gauss curvature is .
Hence is non-constant and monotone non decreasing from the north pole to the equator for , while for it admits a minimum. For , the curvature is maximum on the equator. The limit case corresponds to the Grusin case, for which the curvature is negative everywhere and tends to when tends to .
4.2 Geometric properties
We next present the main properties of the extremal flow for a metric on a two-sphere of revolution satisfying (H1) and (H2) where is smooth, except may be at the equator where it can admit a pole of order one.
For such a family, we consider the smooth Hamiltonian and we restrict extremal curves to the level set . Fixing , the parameterized family of corresponding Hamiltonians described the evolution of the variables as solutions of a mechanical system for which plays the role of potential. For , we get the meridian solutions. Hence, we can assume . Using assumption (H1), the only equilibrium point is for and . This leads to the equator solution in the regular case.
For the remaining trajectories, the level set is sufficient to analyze the behaviors of . Indeed, it is a compact set, symmetric for the two reflections with respect to the -axis and the equator and defined respectively by the two transformations: and . Every trajectory is periodic and oscillates periodically between and . There is also a relation between the period of oscillation and the amplitude , depending upon .
By symmetry, every trajectory is defined by its restriction to a quarter of period, that is the sub-arc starting from the equator and reaching . The trajectory starting from and reaching after passing corresponds to a point rotating on the level set and is chased by a point associated to the trajectory starting from and reaching . They are distinct if . Moreover, using the assumption (H2), we deduce easily that for fixed , the extremals starting from with respectively and intersect with equal length on the antipodal parallel. They are distinct if . The case corresponds for an initial condition not on the equator to tangential arrival and departure at parallels and ; gives the equator solution in the non singular case. For more details see .
The only difference in the singular case is that the equator is not solution, and for trajectories departing from the equator the extremals are always tangential to the meridian, while the first return to the equator can be arbitrarily closed from the initial point.
Finally, another obvious symmetry is a reflectional symmetry with respect to the meridian obtained by changing into .
As a consequence of this analysis, we deduce:
Let be a metric on a two-sphere of revolution, satisfying (H1) and (H2) and smooth except may be at the equator where it can admit a pole of order one.
Then except the meridian and the equator solution in the regular case, every extremal is such that oscillates periodically between two symmetric parallels. The first return mapping to the equator is
where is the corresponding of the extremal.
Assume and then
a) Fixing , changing into gives two distinct extremals with equal length intersecting on the antipodal parallel.
b) Fixing and changing into gives two distinct extremals with equal length intersecting on the opposite meridian.
For the family of metrics which fit in the previous geometric framework, we can be more precise and make a complete analysis. For a fixed value of , the Hamiltonian is:
and corresponds for to the Grusin case. Using , we get:
which can be written
Therefore, and parameterized by arc-length: , one gets the level set .
Hence the integration of the Grusin case gives the general solution, and from the homotopy, the corresponding extremals fit not only in the same geometric framework, but also have the same transcendence.
The family of Hamiltonians admits two first integrals in involution for the Poisson bracket (independent of ) and .
We next outline the integration method in the Grusin case to provide the computation of the first return mapping to the equator obtained in . We have and fixing the level set to , we get:
Taking the positive branch, we must evaluate the following expression
To integrate, we use the relation
to deduce the form of the component of the general solution:
To complete the integration, we write:
and we use the formula:
for with the relation . A straightforward computation then leads to .
For the family of metrics , we have:
In particular, if then on .
This property allows to evaluate conjugate and cut loci for the family of metrics that we next describe .
4.4 Conjugate and cut loci
For (round sphere), the conjugate and cut loci of any point are reduced to the antipodal point.
For , the conjugate locus of a point different from a pole is diffeomorphic to a standard astroid, while the cut locus is a single branch of the antipodal parallel. Both are symmetric with respect to the opposite meridian.
For (Grusin case), the conjugate and cut loci of a point different from a pole and not on the equator are as above. For a point on the equator, the cut locus is the equator minus this point and for the conjugate locus, the cusps on the equator are transformed into folds at this point minus this point.
For the class of metrics , the situation is clear. For the round sphere, all extremals starting from the equator intersect at the same antipodal point and the first return mapping is constant. For , the first return mapping is monotone, and in the singular case as . Since the cut locus of a point of the equator is formed by intersections with the equator of symmetric extremals, in the homotopy, the cut locus is pinched into a point for , while it is stretched into the whole equator in the case .
4.5 Zermelo navigation problem on the two-sphere of revolution
We introduce the following definition for the Zermelo problem.
A Zermelo navigation problem on the two-sphere of revolution is a time-minimum problem of the form:
where the drift representing the current is of the form while and form outside the equator an orthonormal frame for a metric of the form . It is called reflectionaly symmetric with respect to the equator if
(H3) , .
It defines a Finsler geometric problem if for the metric .
According to this classification, we have:
Assume and consider the system (3.2) restricted to the two-sphere:
Then it defines a Zermelo navigation problem on the two-sphere of revolution where the current is , the metric is with a singularity at the equator and the assumptions (H1), (H2) and (H3) are satisfied. The drift can be compensated by a feedback when , which defines a Finsler geometric problem on the sphere minus the equator.
The amplitude of the current is and is maximum in the upper hemisphere for , while it is minimum at the north pole and at the equator. Hence, more generally, we deduce the following proposition.
For , the current can be compensated in the north equator except in a band centered at , hence defining a Finsler geometric problem near the equator and near the north pole.
The controllability analysis is straightforward and is related in the north hemisphere to the scalar equation:
Starting at with , to increase we meet a barrier corresponding to the singularity of the vector field. For instance, if then we have a barrier when .
5 The integrable case of two-level Lindblad equations
5.1 The program
We now proceed to the analysis of the general case of a two-level Lindblad equation. The method is to start from the Grusin case and then to consider perturbations. This program succeeds only if , leading to extremal flows described by a family of 2D-integrable Hamiltonian vector fields.
5.2 The integrable case
We observe that for , the true Hamiltonian simplifies into:
and we immediately deduce:
For , using the coordinate , the Hamiltonian takes the form:
Hence and are cyclic coordinates and , are first integrals of the motion. The system is Liouville integrable.
We have the following geometric interpretation.
For , the Hamiltonian is associated if to the problem of minimization of , while the case corresponds to the maximization of , for .
It is a consequence of the maximum principle. Another point of view is to consider the end-point mapping . If is restricted to the sphere then the solutions of parameterize the singularities of the end-point mapping. The case corresponds to singularities of the end-point mapping for the system restricted to the two-sphere. In the extremum problem of , with fixed time, can be normalized to -1, 0 or 1, while the level sets are . In the extremum problem of time, the Hamiltonian is normalized to 0 or 1 for the minimum case, and 0 or -1 for the maximum one. This is clearly equivalent by homogeneity. Hence, this gives a dual point of view. ∎
In order to indicate the complexity of the problem, we consider first the case of energy, which is equivalent to time from Maupertuis principle, in the Grusin case.
5.2.1 The case of energy
In the normal case, the true Hamiltonian is
We fix the level set to and using the relation , one gets where is the potential:
To integrate, we use:
and we must evaluate an integral of the form:
with . This corresponds to an elliptic integral.
5.2.2 The time-minimum case
In the time-minimum case, the computations of the extremal curves
are more intricate because we cannot reduce the system to a
second-order differential equation. The geometric framework is
however neat because it is associated to a Zermelo navigation
We set and the Hamiltonian is restricted to a level set , where corresponds to the abnormal case and to the normal case. This gives the following relation:
and fixing and , the pair is solution of the system:
Hence leads to:
Using (12), we deduce that is solution of a polynomial equation of degree 2:
The discriminant of this polynomial is:
From (14), we deduce:
Hence the set