Finite controllability of infinite-dimensional quantum systems

IEEE Transactions on Automatic Control

Volumn 55, Issue 8, 2010, Pages 1797-1805

  Bloch, Anthony M ^a   Brockett, Roger W ^b   Rangan, Chitra ^c  

^a UNIVERSITY OF MICHIGAN   (United States)

^b HARVARD UNIVERSITY   (United States)

^c UNIVERSITY OF WINDSOR   (Canada)

Author keywords

Bilinear systems; controllability; infinite dimensional control; Lie algebras; quantum control; Schrodinger equation; trapped ions

Indexed keywords

BILINEAR SYSTEM; BILINEAR SYSTEMS; DIMENSIONAL CONTROL; LIE ALGEBRA; QUANTUM CONTROL;

ALGEBRA; QUANTUM COMPUTERS; QUANTUM ELECTRONICS; QUANTUM OPTICS; SCHRODINGER EQUATION; SPIN DYNAMICS;

CONTROLLABILITY;

EID: 77955391740     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2010.2044273     Document Type: Article

References (44)

1. G. M. Huang, T. J. Tarn, and J. W. Clark, "On the controllability of quantum-mechanical systems," J. Math. Phys., vol.24, no.11, pp. 2608-2618, 1983.
2. M. Shapiro and P. Brumer, "Laser control of product quantum state populations in unimolecular reactions," J. Chem. Phys., vol.84, no.7, pp. 4103-4104, 1986.
3. D. J. Tannor, R. Kosloff, and S. A. Rice, "Coherent pulse sequence induced control of selectivity of reactions-Exact quantum-mechanical calculations," J. Chem. Phys., vol.85, no.10, pp. 5805-5820, 1986.
4. D. J. Tannor and S. A. Rice, "Control of selectivity of chemical-reaction via control of wave packet evolution," J. Chem. Phys., vol.83, no.10, pp. 5013-5018, 1985.
5. A. P. Peirce, M. A. Dahleh, and H. Rabitz, "Optimal-control of quantum-mechanical systems-Existence, numerical approximation, and applications," Phys. Rev. A, vol.37, no.12, pp. 4950-4964, 1988.
6. C. Rangan and P. H. Bucksbaum, "Optimally shaped terahertz pulses for phase retrieval in a Rydberg-atom data register," Phys. Rev. A, vol.6403, no.3, 2001.
7. J. P. Palao and R. Kosloff, "Quantum computing by an optimal control algorithm for unitary transformations," Phys. Rev. Lett., vol.89, no.18, 2002.
8. K. R. Brown, J. Vala, and K. B. Whaley, "Scalable ion trap quantum computation in decoherence-free subspaces with pairwise interactions only," Phys. Rev. A, vol.67, no.1, 2003.
9. N. Khaneja, R. Brockett, and S. J. Glaser, "Time optimal control in spin systems," Phys. Rev. A, vol.6303, no.3, Mar. 2001.
10. E. A. Shapiro, M. Spanner, and M. Y. Ivanov, "Quantum logic approach to wave packet control," Phys. Rev. Lett., vol.91, no.23, 2003.
11. K. Jacobs, "Engineering quantum states of a nanoresonator via a simple auxiliary system," Phys. Rev. Lett., vol.99, no.11, 2007.
12. "See articles in. Implementations of quantum computing," Quantum Inform. Computat., vol.1, 2001.
13. C. K. Law and J. H. Eberly, "Arbitrary control of a quantum electromagnetic field," Phys. Rev. Lett., vol.76, no.7, pp. 1055-1058, 1996.
14. B. Kneer and C. K. Law, "Preparation of arbitrary entangled quantum states of a trapped ion," Phys. Rev. A, vol.57, no.3, pp. 2096-2104, 1998.
15. A. Ben-Kish et al., "Experimental demonstration of a technique to generate arbitrary quantum superposition states of a harmonically bound spin-1/2 particle," Phys. Rev. Lett., vol.90, no.3, 2003.
16. R. W. Brockett, C. Rangan, and A. M. Bloch, "The controllability of infinite quantum systems," in Proc. 42nd IEEE Conf. Decision Control, 2003, pp. 428-433.
17. C. Rangan, A. M. Bloch, C. Monroe, and P. H. Bucksbaum, "Control of trapped-ion quantum states with optical pulses," Phys. Rev. Lett., vol.92, no.11, 2004.
18. C. Rangan and A. M. Bloch, "Control of finite-dimensional quantum systems: Application to a spin-1/2 particle coupled with a finite quantum harmonic oscillator," J. Math. Phys., vol.46, no.3, 2005.
19. R. W. Brockett, "System theory on group manifolds and coset spaces," SIAM J. Control, vol.10, no.2, p. 265, 1972.
20. R. W. Brockett, "Lie theory and control-systems defined on spheres," SIAM J. Appl. Math., vol.25, no.2, pp. 213-225, 1973.
21. V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rabitz, and A. Peirce, "Controllability of molecular-systems," Phys. Rev. A, vol.51, no.2, pp. 960-966, 1995.
22. F. Albertini and D. D'Alessandro, "Notions of controllability for bilinear multilevel quantum systems," IEEE Trans. Autom. Control, vol.48, no.8, pp. 1399-1403, Aug. 2003.
23. E. Zuazua, "Remarks on the controllability of the Schrödinger equation," in Proc. CRM, Montreal, QC, Canada, 2000, pp. 1-17.
24. G. Turinici and H. Rabitz, "Wavefunction controllability in quantum systems," J. Phys. A, vol.36, pp. 2565-2576, 2003.
25. J. M. Ball, J. E. Marsden, and M. Slemrod, "Controllability for distributed bilinear-systems," SIAM J. Control Optim., vol.20, no.4, pp. 575-597, 1982.
26. K. Beauchard, "Local controllability of a 1-D Schrödinger equation," J. Math. Pures et Appl., vol.84, pp. 851-956, 2005.
27. R. Illner, H. Lange, and H. Teismann, "Limitations on the control of Schrödinger equations," ESAIM: COCV, vol.12, pp. 615-635, 2006.
28. G. Turinici, "Controllable quantities for bilinear quantum systems," in Proc. 39th IEEE Conf. Decision Control, 2000, pp. 1364-1369.
29. E. A. Shapiro, M. Y. Ivanov, and Y. Billig, "Coarse-grained controllability of wavepackets by free evolution and phase shifts," J. Chem. Phys., vol.120, no.21, pp. 9925-9933, 2004.
30. W. Karwowski and R. V. Mendes, "Quantum control in infinite dimensions," Phys. Lett. A, vol.322, no.5-6, pp. 282-285, 2004.
31. C. H. Lan, T. J. Tarn, Q. S. Chi, and J. W. Clark, "Analytic controllability of time-dependent quantum control systems," J. Math. Phys., vol.46, no.5, 2005.
32. R. Adami and U. Boscain, "Controllability of the schroedinger equation via intersection of eigenvalues," in Proc. 44rd IEEE Conf. Decision Control, 2005, [CD ROM].
33. R. B.Wu, T. J. Tarn, and C. W. Li, "Smooth controllability of infinitedimensional quantum-mechanical systems," Phys. Rev. A, vol.73, no.1, 2006.
34. T. Chambrion, P. Mason, M. Sigalotti, and U. Boscain, "Controllability of the discrete-spectrum schroedinger equation driven by an external field," Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, no.1, pp. 329-349, 2009.
35. L. I. Schiff, Quantum Mechanics. New York: McGraw-Hill, 1968.
36. M. Mirrahimi and P. Rouchon, "Controllability of quantum harmonic oscillators," IEEE Trans. Autom. Control, vol.49, no.5, pp. 745-747, May 2004.
37. B. W. Shore, Theory of Coherent Atomic Excitation, Volume 2, Multilevel Atoms and Incoherence. New York: Wiley-VCH, 1990.
38. S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett., vol.82, no.8, pp. 1784-1787, 1999.
39. K. Kime, "Control of transition-probabilities of the quantum-mechanical harmonic-oscillator," Appl. Math. Lett., vol.6, no.3, pp. 11-15, 1993.
40. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, "Experimental issues in coherent quantum-state manipulation of trapped atomic ions," J. Res. Nat. Inst. Standards Technol., vol.103, no.3, pp. 259-328, 1998.
41. An Expanded Version of This Paper With Complete Derivations and More Examples can be Found at arXiv:quant-ph/0608075v2 2009.
42. E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to beam maser," Proc. IEEE, vol.51, no.1, p. 89, Jan. 1963.
43. G. Turinici, "Exact controllability for the population of the eigenstates in bilinear quantum systems," Comptes Rendus de l Academie des Sciences Serie I-Mathematique, vol.330, no.4, pp. 327-332, 2000.
44. L. H. Pedersen and C. Rangan, "Controllability and universal threequbit quantum computation with trapped electron states," Quantum Inform. Processing, vol.7, pp. 33-42, 2008.