Abstract
This paper presents sufficient and necessary conditions for the propagator controllability of a class of infinite-dimensional quantum systems with SU(1, 1) dynamical symmetry through the isomorphic mapping to the non-unitary representation of SU(1, 1). We prove that the elliptic condition of the total Hamiltonian is both necessary and sufficient for the controllability and strong controllability. The obtained results can be also extended to control systems with SO(2, 1) dynamical symmetry.
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References
Dong D Y and Petersen I R, Controllability of quantum systems with switching control, International Journal of Control, 2011, 84(1): 37–46.
Boscain U, Gauthier J P, Rossi F, et al., Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Communications in Mathematical Physics, 2015, 333(3): 1225–1239.
Indra K, Gunther D, and Uwe H, Controllability aspects of quantum dynamics: A unified approach for closed and open systems, IEEE Transactions on Automatic Control, 2012, 57(8): 1984–1996.
Bloch A M, Brockett R W, and Rangan C, Finite controllability of infinite-dimensional quantum systems, IEEE Transactions on Automatic Control, 2010, 55(8): 1797–1805.
Altafini C, Controllability of quantum mechanical systems by root space decomposition of su(N), J. Math. Phys., 2002, 43: 2051–2062.
Tarn T J, Clark J W, and Lucarelli D G, Controllability of quantum mechanical systems with continuous spectra, Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, 2000, 943–948.
Jurdjevic V and Sussman H J, Control systems on Lie groups, J. Diff. Eqns., 1972, 12: 313–329.
Sussmann H J, Controllability of nonlinear systems, J. Diff. Eqns., 1972, 12: 95–116.
Brockett R W, System theory on group manifolds and coset spaces, SIAM J. Control, 1972, 10(2): 265–284.
Sussmann H J, Lie brackets, Real Analyticity and Geometric Control, Eds. by Brockett R W, Millman R S, and Sussmann H J, Differential Geometric Control Theory, Birkhauser, Boston, 1982, 1–116.
Jurdjevic V and Kupka I, Control systems subordinated to a group action: Accessibility, J. Diff. Eqs., 1981, 39: 186–211.
Jurdjevic V and Kupka I, Control systems on semisimple Lie groups and their homogeneous spaces, Ann. Institute Fourier, 1981, 31: 151–179.
Assoudi R E, Gauthier J P, and Kupka I, On subsemigroups of semisimple Lie groups, Ann. lnst. Henri Poincaré, Anal. Non Lineaire, 1996, 13: 117–133.
Huang G M, Tarn T J, and Clark J W, On the controllability of quantum-mechanical systems, J. Math. Phys., 1983, 24: 2608–2618.
Ramakrishna V, Salapaka M V, Dahleh M, et al., Controllability of molecular systems, Phys. Rev. A, 1995, 51(2): 960–966.
D’Alessandro D, Topological properties of reachable sets and the control of quantum bits, Systems & Control Letters, 2000, 41: 13–221.
Schirmer S G, Fu H, and Solomon A I, Complete controllability of quantum systems, Phys. Rev. A, 2001, 63: 063410.
Lan C H, Tarn T J, Chi Q S, et al., Controllability of time-dependent quantum control systems, J. Math. Phys., 2005, 46: 052102.
Wu R B, Tarn T J, and Li C W, Smoth controlllability of infinite-dimensional quantum-mechanical system, Phys. Rev. A, 2006, 73: 012719.
Puri R R, Mathematical Methods of Quantum Optics, Springer, Berlin, 2001.
Gerry C C and Vrscay E R, Dynamics of pusled SU(1, 1) coherent states, Phys. Rev. A, 1989, 39: 5717–5724.
Gortel Z W and Turski L A, Classical dynamics for a class of SU(1, 1) Hamiltonians, Phys. Rev. A, 1991, 43: 3221–3226.
Bose S K, Dynamical algebra of spin waves in localised-spin models, J. Phys. A, 1985, 18: 903–922.
Agarwal G S and Banerji J, Reconstruction of SU(1, 1) States, Phys. Rev. A, 2001, 64: 023815.
Walls D F, Squeezed States of Light, Nature, 1983, 306: 141–146.
Gerry C C, Dynamics of SU(1, 1) Coherent States, Phys. Rev. A, 1985, 31: 2721–2723.
Perelomov P, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986.
Dong W B, Wu R B, Wu J W, et al., Optimal control of quantum systems with SU(1, 1) dynamical symmetry, Control Theory Tech., 2015, 13: 211–220.
Wu J W, Li C W, Wu R B, et al., Quantum Control by Decomposition of SU(1, 1), J. Phys. A, 2006, 39: 13531–13551.
Vilenkin N J and Klimyk A U, Representation of Lie Groups and Special Functions, Vol 1: Simplest Lie Groups, Special Functions and Integral Transforms, Kluwer Academic Publishers, Boston, 1991.
Jurdjevic V, Geometric Control Theory, Cambridge University Press, 1997.
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This research was supported by the National Natural Science Foundation of China under Grant Nos. 61803357, 61833010, 61773232, 61622306 and 11674194, the National Key R&D Program of China under Grant Nos. 2018YFA0306703 and 2017YFA0304304, the Tsinghua University Initiative Scientific Research Program, and the Tsinghua National Laboratory for Information Science and Technology Cross-discipline Foundation.
This paper was recommended for publication by Editor SUN Jian.
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Wu, J., Wu, R., Zhang, J. et al. Controllability of Quantum Systems with SU(1, 1) Dynamical Symmetry. J Syst Sci Complex 34, 827–842 (2021). https://doi.org/10.1007/s11424-020-9259-9
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DOI: https://doi.org/10.1007/s11424-020-9259-9