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### The energy minimization problem for two-level dissipative quantum systems

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Affiliations:
1 Institut de Mathématiques de Bourgogne, UMR CNRS 5584, BP 47870, 21078 Dijon, France
2 IRIT, ENSEEIHT, UMR CNRS 5055, 31062 Toulouse, France
3 Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France
a) Electronic mail: bernard.bonnard@u-bourgogne.fr.
b) Electronic mail: dominique.sugny@u-bourgogne.fr.
J. Math. Phys. 51, 092705 (2010)
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### References

• B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny
• Source: J. Math. Phys. 51, 092705 ( 2010 );
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http://aip.metastore.ingenta.com/content/aip/journal/jmp/51/9/10.1063/1.3479390
View: Figures

## Figures

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FIG. 1.

Phase portraits in the case for (left) and (right). In the left panel, numbers 1, 2, 3, 4, and 5 are, respectively, associated with , , , , and . In the right panel, they correspond, respectively, to , , , , and .

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FIG. 2.

Phase portraits in the case for (top), (middle), and for (bottom).

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FIG. 3.

Behavior near the origin of a family of orbits (see the text).

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FIG. 4.

Foliation of (a) and level sets of the value function (b). (Bottom panel) Conjugate point analysis. Numerical values are taken to be , from top to bottom.

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FIG. 5.

(Top) Generic microlocal case. (Bottom) The Grushin case.

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FIG. 6.

(Top) Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different dissipation parameters. Numerical values are, respectively, taken to be , 1.24, and 2.02 for the red (dark gray), green (light gray), and blue (black) trajectories.

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FIG. 7.

Plot of the different optimal trajectories (top) and of the corresponding control fields (bottom) for different control durations. Numerical values are, respectively, taken to be , 1.5, and 2 for the red (dark gray), green (light gray), and blue (black) trajectories. The value refers to the time-minimum solution for the same dissipation parameters and a maximum normalized amplitude of the control field of . The black crosses indicate the positions of the different conjugate points. The middle panel is a zoom of the top figure near the origin.

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FIG. 8.

(a) Plot of the potential as a function of . (b) Plot of four extremals corresponding to , 0, 1, and . (c) Same as (b) but up to the first conjugate point. Other numerical values are taken to be , , , , and . The extremals associated with and 1 are short periodic orbits with an energy equal to 5.5, while extremals with initial adjoint states and 2 are long periodic orbits with an energy equal to 6.5. The horizontal dashed lines indicate the positions of the parallel of equation and the antipodal one of equation where short and long periodic orbits, respectively, intersect with the same time.

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FIG. 9.

Conjugate loci, spheres, and wave fronts for the Grushin (top) and the non-Grushin case (bottom).

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FIG. 10.

(a) Evolution of the angle as a function of the angle for , , and . Initial values are taken to be , , , and , 0, and 1. (b) Evolution of the radial coordinate as a function of time. [(c) and (d)] Plot of the two optimal control fields and as a function of time for .

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FIG. 11.

The same as Fig. 10 but for .

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FIG. 12.

The same as Fig. 10 but the extremals are plotted up to the first conjugate point.

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FIG. 13.

The same as Fig. 12 but for .

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FIG. 14.

Evolution as a function of the parameter of the initial adjoint states , , and for the Newton-type continuation (solid line) and the smooth-type one (dashed line). The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

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FIG. 15.

Three extremal trajectories solutions of the smooth continuation method for , 2.5, and 3. The black dot indicates the position of the target state.

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FIG. 16.

The same as Fig. 14 but for a continuation in . The other dissipative parameters are and and the control duration is equal to 0.5. The coordinates of the initial point are equal to , , , , , and for . The target state is (, , ).

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FIG. 17.

Three extremal trajectories solutions of the smooth continuation method for , 0, and 0.1. The black dot indicates the position of the target state.

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FIG. 18.

Evolution of the continuation parameters as a function of the number of steps for the discrete method (solid line) and the smooth one (dashed line).

/content/aip/journal/jmp/51/9/10.1063/1.3479390
2010-09-15
2013-12-06

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