April  2020, 13(4): 1061-1073. doi: 10.3934/dcdss.2020063

Time-minimum control of quantum purity for 2-level Lindblad equations

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M), Calle Nicolás Cabrera 13-15, Campus Cantoblanco, 28409 Madrid, Spain

3. 

Dept. of Physics, University of Windsor, Ontario, N9B 3P4, Canada

* Corresponding author: Anthony Bloch

Received  February 2018 Revised  August 2018 Published  April 2019

Fund Project: A. Bloch and W. Clark were supported by NSF grant DMS-1613819 and A. Bloch was also supported by AFSOR grant FA9550-18-1-0028. L. Colombo was partially supported by MINECO (Spain) grant MTM2016-76072-P and Juan de la Cierva Incorporación Fellowship.

We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.

The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

Citation: William Clark, Anthony Bloch, Leonardo Colombo, Patrick Rooney. Time-minimum control of quantum purity for 2-level Lindblad equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1061-1073. doi: 10.3934/dcdss.2020063
References:
[1]

C. Altafini, Controllability properties for finite-dimensional quantum Markovian master equations, J. Math. Phys., 44 (2002), 2357-2372.  doi: 10.1063/1.1571221.

[2]

A. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[3]

B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Mathématiques & Appl., 40 Springer-Verlag, Berlin, 2003.

[4]

B. BonnardM. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case, IEEE Transactions on Automatic Control, 54 (2009), 2598-2610.  doi: 10.1109/TAC.2009.2031212.

[5]

B. Bonnard and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The integrable case, SIAM J. Control Optim., 48 (2009), 1289-1308.  doi: 10.1137/080717043.

[6]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer-Verlag, Berlin, 2004.

[7]

U. BoscainM. Caponigro and M. Sigalotti, Multi-input Schrödinger equation: Controllability, tracking, and application to the quantum angular momentum, J. of Diff. Eqns., 256 (2014), 3524-3551.  doi: 10.1016/j.jde.2014.02.004.

[8] H. Breuer and F. Pertuccione, The Theory of Open Quantum Systems, Oxford University Press, New York, 2002. 
[9]

R. W. Brockett, On the reachable set for bilinear systems, Variable Structure Systems with Application to Economics and Biology (Proc. Second U.S.-Italy Sem., 1974), 1975, 54-63.

[10]

G. CharlotJ. P. GauthierU. BoscainS. Guérin and H. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., 43 (2002), 2107-2132.  doi: 10.1063/1.1465516.

[11]

W. Clark, A. Bloch, L. Colombo and P. Rooney, Optimal control of quantum purity for n = 2 systems, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, 2017, 1317-1322. doi: 10.1109/CDC.2017.8263837.

[12]

R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two dimensions, Clarendon, Oxford, 1987.

[13]

S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K$\ddot{o}$ckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr$\ddot{u}$ggen, D. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, Eur. Phys. J. D, 69 (2015), 279. doi: 10.1140/epjd/e2015-60464-1.

[14]

J. D. Hoffman and S. Frankel, Numerical Methods for Engineers and Scientists, Second Edition, Taylor & Francis, 2001.

[15]

J. Jover Galtier, Open Quantum Systems: Geometric Description, Dynamics and Control, Ph.D thesis, Universidad de Zaragoza, Spain. 2017.

[16]

N. Khaneja, R. Brockett and S. Glaser, Time optimal control of spin systems, Phys. Rev. A, 63 (2001), 032308.

[17]

N. Khaneja, S. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Coherence transfer and quantum gates, Phys. Rev. A, 65 (2002), part A, 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[18]

D. E. Kirk, Optimal Control Theory: An Introduction, Dover Books on Electrical Engineering Series. Dover Publications, 2004.

[19]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular Extremals for the Time-Optimal Control of Dissipative Spin 1/2 Particles, Phys. Rev Lett., 104 (2010), 083001.

[20]

G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119-130.  doi: 10.1007/BF01608499.

[21]

S. Ramakrishna and T. Seideman, Intense laser alignment in dissipative media as a route of solvent dynamics, Phys. Rev Lett., 95 (2005), 113001. doi: 10.1103/PhysRevLett.95.113001.

[22]

C. Rangan and P. Bucksbaum, Optimality shaped terahertz pulses for phase retrieval in a Rydberg-atom data registrer, Phys. Rev. A, 89 (2002), 188301.

[23]

P. Rooney, A. Bloch and C. Rangan, Flag-based control of quantum purity for n = 2 systems, Phys. Rev. A, 93 (2016), 063424.

[24]

P. Rooney, A. Bloch and C. Rangan, Projector-based control of orbit dynamics in quantum lindblad systems (2017, to appear in the IEEE Trans. Aut. Control), Preprint, arXiv: 1201.0399v1.

[25]

P. Rooney, A. Bloch and C. Rangan, Decoherence control and purification of two-dimensional quantum density matrices under Lindblad dissipation, Preprint, arXiv: 1201.0399v1.

[26]

D. Sugny, C. Kontz and H. R. Jauslin, Time-optimal control of two-level dissipative quantum system, Phys. Rev. A, 76 (2007), 023419. doi: 10.1103/PhysRevA.76.023419.

[27]

D. Tannor and A. Bartana, On the interplay of control fields and spontaneous emission in laser cooling, J. Phys Chem A, 103 (1999), 10359-10363.  doi: 10.1021/jp992544x.

[28]

Th. Viellard, F. Chaussard, F. Billard, D. Sugny, O. Faucher, S. Ivanov, J.-M. Hartmann, C. Boulet and B. Lavorel, Field-free molecular alignment for probing collisional relaxation dynamics, Phys. Rev. A, 87 (2013), 023409. doi: 10.1103/PhysRevA.87.023409.

show all references

References:
[1]

C. Altafini, Controllability properties for finite-dimensional quantum Markovian master equations, J. Math. Phys., 44 (2002), 2357-2372.  doi: 10.1063/1.1571221.

[2]

A. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[3]

B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Mathématiques & Appl., 40 Springer-Verlag, Berlin, 2003.

[4]

B. BonnardM. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case, IEEE Transactions on Automatic Control, 54 (2009), 2598-2610.  doi: 10.1109/TAC.2009.2031212.

[5]

B. Bonnard and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The integrable case, SIAM J. Control Optim., 48 (2009), 1289-1308.  doi: 10.1137/080717043.

[6]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer-Verlag, Berlin, 2004.

[7]

U. BoscainM. Caponigro and M. Sigalotti, Multi-input Schrödinger equation: Controllability, tracking, and application to the quantum angular momentum, J. of Diff. Eqns., 256 (2014), 3524-3551.  doi: 10.1016/j.jde.2014.02.004.

[8] H. Breuer and F. Pertuccione, The Theory of Open Quantum Systems, Oxford University Press, New York, 2002. 
[9]

R. W. Brockett, On the reachable set for bilinear systems, Variable Structure Systems with Application to Economics and Biology (Proc. Second U.S.-Italy Sem., 1974), 1975, 54-63.

[10]

G. CharlotJ. P. GauthierU. BoscainS. Guérin and H. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., 43 (2002), 2107-2132.  doi: 10.1063/1.1465516.

[11]

W. Clark, A. Bloch, L. Colombo and P. Rooney, Optimal control of quantum purity for n = 2 systems, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, 2017, 1317-1322. doi: 10.1109/CDC.2017.8263837.

[12]

R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two dimensions, Clarendon, Oxford, 1987.

[13]

S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K$\ddot{o}$ckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr$\ddot{u}$ggen, D. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, Eur. Phys. J. D, 69 (2015), 279. doi: 10.1140/epjd/e2015-60464-1.

[14]

J. D. Hoffman and S. Frankel, Numerical Methods for Engineers and Scientists, Second Edition, Taylor & Francis, 2001.

[15]

J. Jover Galtier, Open Quantum Systems: Geometric Description, Dynamics and Control, Ph.D thesis, Universidad de Zaragoza, Spain. 2017.

[16]

N. Khaneja, R. Brockett and S. Glaser, Time optimal control of spin systems, Phys. Rev. A, 63 (2001), 032308.

[17]

N. Khaneja, S. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Coherence transfer and quantum gates, Phys. Rev. A, 65 (2002), part A, 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[18]

D. E. Kirk, Optimal Control Theory: An Introduction, Dover Books on Electrical Engineering Series. Dover Publications, 2004.

[19]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular Extremals for the Time-Optimal Control of Dissipative Spin 1/2 Particles, Phys. Rev Lett., 104 (2010), 083001.

[20]

G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119-130.  doi: 10.1007/BF01608499.

[21]

S. Ramakrishna and T. Seideman, Intense laser alignment in dissipative media as a route of solvent dynamics, Phys. Rev Lett., 95 (2005), 113001. doi: 10.1103/PhysRevLett.95.113001.

[22]

C. Rangan and P. Bucksbaum, Optimality shaped terahertz pulses for phase retrieval in a Rydberg-atom data registrer, Phys. Rev. A, 89 (2002), 188301.

[23]

P. Rooney, A. Bloch and C. Rangan, Flag-based control of quantum purity for n = 2 systems, Phys. Rev. A, 93 (2016), 063424.

[24]

P. Rooney, A. Bloch and C. Rangan, Projector-based control of orbit dynamics in quantum lindblad systems (2017, to appear in the IEEE Trans. Aut. Control), Preprint, arXiv: 1201.0399v1.

[25]

P. Rooney, A. Bloch and C. Rangan, Decoherence control and purification of two-dimensional quantum density matrices under Lindblad dissipation, Preprint, arXiv: 1201.0399v1.

[26]

D. Sugny, C. Kontz and H. R. Jauslin, Time-optimal control of two-level dissipative quantum system, Phys. Rev. A, 76 (2007), 023419. doi: 10.1103/PhysRevA.76.023419.

[27]

D. Tannor and A. Bartana, On the interplay of control fields and spontaneous emission in laser cooling, J. Phys Chem A, 103 (1999), 10359-10363.  doi: 10.1021/jp992544x.

[28]

Th. Viellard, F. Chaussard, F. Billard, D. Sugny, O. Faucher, S. Ivanov, J.-M. Hartmann, C. Boulet and B. Lavorel, Field-free molecular alignment for probing collisional relaxation dynamics, Phys. Rev. A, 87 (2013), 023409. doi: 10.1103/PhysRevA.87.023409.

Figure 1.  Trajectory and controls of the $7^{th}$ order curve. The black ellipse is the escape chimney
Figure 2.  Left: Example where $ b = [0;0] $, $ \alpha = -3 $, and $ \beta = -0.6 $. Right: Example where $ b = [0;0] $, $ \alpha = -0.8 $, and $ \beta = -0.6 $
Figure 3.  Example where $ b = [-2;-1] $, $ \alpha = -4 $, and $ \beta = -3 $. The red curves show the trajectory when $ u = -1 $ and blue when $ u = 1 $
Table 1.  Numerical results from time-minimal controls with solutions of various orders
MTimeEnergy
11.93727.5830
31.93668.6873
51.93611.6368
71.93591.3765
MTimeEnergy
11.93727.5830
31.93668.6873
51.93611.6368
71.93591.3765
Table 2.  Switching times for the example in Figure 3
Initial controlTime until first switchTime between first and second switches
$u = 1$2.19430.4685
$u = -1$1.15320.4905
Initial controlTime until first switchTime between first and second switches
$u = 1$2.19430.4685
$u = -1$1.15320.4905
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