Abstract
We derive symplectically invariant uncertainty relations for a set of canonically conjugated variables. The uncertainty relations obtained are multimode analogs of the Robertson–Schrödinger inequalities. Our uncertainty relations are equivalent to the necessary and sufficient conditions for a matrix to be a correlation matrix of some quantum state, obtained by R. Simon and coauthors. The advantage of our inequalities, compared to that suggested by Simon, consists in its simplicity and more obvious symplectic invariance. We derive our inequalities for the case of a two-mode system in explicit form. Particular cases of small and large degrees of correlation between the first and second modes are analyzed in detail.
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References
H. P. Robertson, Phys. Rev., 34, 163 (1929).
E. Schrödinger, Collected Papers on Quantum Mechanics, [in Russian] Nauka, Moscow (1976).
E. Schrödinger, Collected Papers, Friedr. Vieweg & Sohn Verlag, Germany (1984).
V. V. Dodonov and V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute [in Russian], Nauka, Moscow (1987), Vol. 183 [Translated into English by Nova Science Publishers, Commack, New York (1989)].
V. V. Dodonov, M. A. Man’ko, V. I. Man’ko, and A. Vourdas, J. Russ. Laser Res., 28, 404 (2007).
N. V. Karelin and A. M. Lazaruk, Teor. Mat. Fiz., 117, 427 (1998) [Theor. Math. Phys., 117, 1447 (1998)].
M. U. Karelin and A. M. Lazaruk, J. Phys. A: Math. Gen., 33, 6807 (2000).
D. A. Trifonov and S. G. Donev, J. Phys. A: Math. Gen., 31, 8041 (1998).
D. A. Trifonov, J. Phys. A: Math. Gen., 30, 5941 (1998).
R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A, 49, 1567 (1994).
E. C. G. Sudarshan, C. B. Chiu, and G. Bhamathi, Phys. Rev. A, 52, 43 (1995).
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1978).
P. S. Alekseev and F. V. Moroseev, J. Exp. Theor. Phys., 108, 571 (2009).
R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A, 37, 3028 (1988).
A. Eckdtein, A. Christ, P. J. Mosley, and C. Silberhorn, Phys. Rev. Lett., 106, 013603 (2011).
V. D’Auria, S. Fornaro, A. Porizio, et al., Phys. Rev. Lett., 102, 020502 (2009).
S. Pirandola, A. Serafini, and S. Lloyd, Phys. Rev. A, 79, 052327 (2009).
G. Giedke, M. M. Wolf, O. Kruger, et al., Phys. Rev. Lett., 91, 107901 (2003).
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Alekseev, P.S., Moroseev, F.V. A new form of symplectically invariant multimode uncertainty relations. J Russ Laser Res 32, 311–316 (2011). https://doi.org/10.1007/s10946-011-9218-y
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DOI: https://doi.org/10.1007/s10946-011-9218-y