Abstract

Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.

© 2000 Optical Society of America

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2000 (1)

D. A. Trifonov, “State extended uncertainty relations,” J. Phys. A 33, L299–L304 (2000), E-print quant-ph/0005086, http://arxiv.org/abs/quant-ph/yymmnnn .
[CrossRef]

1999 (4)

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, A. Wünsche, “Energy-sensitive and ‘classical-like’ distances between quantum states,” Phys. Scr. 59, 81–89 (1999); D. A. Trifonov, S. G. Donev, “Polarized distances between quantum states and observables,” E-print quant-ph/0005087, http://arxiv.org/abs/quant-ph/yymmnnn .
[CrossRef]

S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
[CrossRef]

D. A. Trifonov, “Exact solution for the general nonstationary oscillator with a singular perturbation,” J. Phys. A 32, 3649–3661 (1999).
[CrossRef]

G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999).
[CrossRef]

1998 (3)

D. A. Trifonov, “On the squeezed states for n observables,” Phys. Scr. 58, 246–255 (1998).
[CrossRef]

D. A. Trifonov, S. G. Donev, “Characteristic uncertainty relations,” J. Phys. A 31, 8041–8047 (1998).
[CrossRef]

D. A. Trifonov, “Barut–Girardello coherent states for u(p, q) and sp(N, R) and their macroscopic superpositions,” J. Phys. A 31, 5673–5696 (1998).
[CrossRef]

1997 (5)

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

S. Mancini, “Even and odd nonlinear coherent states,” Phys. Lett. A 233, 291–296 (1997); S. Sivakumar, “Generation of even and odd nonlinear coherent states,” E-print quant-ph/9902054, http://arxiv.org/abs/quant-ph/yymmnnn ; B. Roy, P. Roy, “Phase properties of even and odd nonlinear coherent states,” Phys. Lett. A 257, 264–268 (1999); “Time dependent nonclassical properties of even/odd nonlinear coherent states,” Phys. Lett. A 263, 48–52 (1999).
[CrossRef]

K. Fujii, K. Funahashi, “Extension of the Barut–Girardello coherent state and path integral,” J. Math. Phys. 38, 4422–4434 (1997).
[CrossRef]

C. Brif, “SU(2) and SU(1, 1) algebra eigenstates: a unified analytic approach to coherent and intelligent states,” Int. J. Theor. Phys. 36, 1651–1682 (1997).
[CrossRef]

D. A. Trifonov, “Robertson intelligent states,” J. Phys. A 30, 5941–5957 (1997).
[CrossRef]

1996 (5)

C. Brif, “Two-photon algebra eigenstates. A unified approach to squeezing,” Ann. Phys. (N.Y.) 251, 180–207 (1996).
[CrossRef]

R. J. McDermott, A. I. Solomon, “Squeezed states parametrized by elements of noncommutative algebras,” Czech. J. Phys. 46, 235–241 (1996).
[CrossRef]

S. L. Braunstein, C. M. Caves, G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. (N.Y.) 247, 135–175 (1996).
[CrossRef]

C. Brif, A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A 54, 4505–4518 (1996).
[CrossRef] [PubMed]

A. Luis, J. Perina, “SU(2) coherent states in parametric down-conversion,” Phys. Rev. A 53, 1886–1893 (1996).
[CrossRef] [PubMed]

1995 (2)

A. Kempf, G. Mangano, R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D 52, 1108–1118 (1995).
[CrossRef]

E. S. G. Sudarshan, C. B. Chiu, G. Bhamathi, “Generalized uncertainty relations and characteristic invariants for multimode states,” Phys. Rev. A 52, 43–54 (1995).
[CrossRef] [PubMed]

1994 (3)

D. A. Trifonov, “Generalized intelligent states and squeezing,” J. Math. Phys. 35, 2297–2308 (1994); “Generalized intelligent states and SU(1, 1) and SU(2) squeezing,” Preprint INRNE-TH-93/4 (May1993) [quant-ph/0001028]; available on request from D. A. Trifonov.
[CrossRef]

R. R. Puri, “Minimum uncertainty states for noncanonical operators,” Phys. Rev. A 49, 2178–2180 (1994); R. R. Puri, G. S. Agarwal, “SU(1, 1) coherent states defined via a minimum-uncertainty-product and an equality of quadrature variances,” Phys. Rev. A 53, 1786–1790 (1996); R. Simon, N. Mukunda, “Moments of the Wigner distribution and a generalized uncertainty principle,” E-print quant-ph/9708037, http://arxiv.org/abs/quant-ph/yymmnnn .
[CrossRef] [PubMed]

N. A. Ansari, V. I. Man’ko, “Photon statistics of multimode even and odd coherent light,” Phys. Rev. A 50, 1942–1945 (1994); V. V. Dodonov, V. I. Man’ko, D. E. Nikonov, “Even and odd coherent states for multimode parametric systems,” Phys. Rev. A 51, 3328–3336 (1995).
[CrossRef] [PubMed]

1991 (2)

J. A. Bergou, M. Hillery, D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991); M. M. Nieto, D. R. Truax, “Squeezed states for general systems,” Phys. Rev. Lett. 71, 2843–2846 (1993).
[CrossRef] [PubMed]

D. A. Trifonov, “On the stable evolution of squeezed and correlated states,” J. Sov. Laser Res. 12, 414–420 (1991); “Completeness and geometry of Schrödinger minimum uncertainty states,” J. Math. Phys. 34, 100–110 (1993).
[CrossRef]

1990 (2)

W.-M. Zhang, D. H. Feng, R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–924 (1990); S. Tareque Ali, J.-P. Antoine, J.-P. Gazeau, U. A. Mueler, “Coherent states and their generalizations: a mathematical overview,” Rev. Math. Phys. 7, 1013–1104 (1995).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, O. V. Man’ko, “Nonstationary quantum oscillator,” Proc. P. N. Lebedev Phys. Inst. 191, 171–244 (1990); A. K. Angelow, “Light propagation in nonlinear waveguide and classical two-dimensional oscillator,” Physica A 256, 485–498 (1998).
[CrossRef]

1987 (4)

J. Katriel, A. I. Solomon, G. D’Ariano, M. Rasetti, “Multiphoton squeezed states,” J. Opt. Soc. Am. B 4, 1728–1735 (1987).
[CrossRef]

R. Loudon, P. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 37, 3868–3880 (1987); X. Ma, W. Rhodes, “Multimode squeeze operators and squeezed states,” Phys. Rev. A 41, 4624–4631 (1990); V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, “Multidimensional Hermite polynomials and photon distribution for polymode mixed light,” Phys. Rev. A 50, 813–817 (1994).
[CrossRef] [PubMed]

V. V. Dodonov, V. I. Man’ko, “Generalizations of the uncertainty relations in quantum mechanics,” Proc. P. N. Lebedev Phys. Inst. 183, 3–70 (1987).

1985 (1)

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981); “Defense of the standard quantum limit for free-mass position,” Phys. Rev. Lett. 54, 2465–2468 (1985).
[CrossRef] [PubMed]

1980 (1)

V. V. Dodonov, E. V. Kurmyshev, V. I. Man’ko, “Generalized uncertainty relation and correlated coherent states,” Phys. Lett. A 79, 150–152 (1980); V. V. Dodonov, V. I. Man’ko, “Invariants and correlated states of nonstationary quantum systems,” in Proc. P. N. Lebedev Phys. Inst. 183, 71–181 (1987).
[CrossRef]

1979 (1)

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detection,” Phys. Rev. D 19, 1669–1679 (1979).
[CrossRef]

1976 (2)

H. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

C. Aragone, E. Chalbaud, S. Salamo, “On intelligent spin states,” J. Math. Phys. 17, 1963–1971 (1976).
[CrossRef]

1975 (1)

D. A. Trifonov, “On coherent states of quantum systems and uncertainty relations,” Bulg. J. Phys. 2, 303–311 (1975).

1974 (1)

V. V. Dodonov, I. A. Malkin, V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator,” Physica (Amsterdam) 72, 597–618 (1974).
[CrossRef]

1972 (1)

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972).
[CrossRef]

1971 (2)

A. O. Barut, L. Girardello, “New ‘coherent’ states associated with noncompact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

J. M. Radcliffe, “Some properties of coherent spin states,” J. Phys. A 4, 313–323 (1971); F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

1970 (2)

D. A. Stoler, “Equivalent classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970).
[CrossRef]

I. A. Malkin, V. I. Man’ko, “Coherent states and excitation of n-dimensional nonstationary forced oscillator,” Phys. Lett. A 32, 243–244 (1970); A. Holz, “N-dimensional anisotropic oscillator in a time-dependent homogeneous electromagnetic field,” Lett. N. Cimento A 4, 1319–1323 (1970); I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Dynamical symmetry of nonstationary systems,” Nuovo Cimento A 4, 773–793 (1971).
[CrossRef]

1969 (1)

I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Invariants and evolution of coherent states of charged particles in a time dependent magnetic field,” Phys. Lett. A 30, 414–414 (1969); “Coherent states and transition probabilities in a time dependent electromagnetic field,” Phys. Rev. D 2, 1371–1385 (1970).
[CrossRef]

1966 (1)

M. M. Miller, E. A. Mishkin, “Characteristic states of the electromagnetic radiation field,” Phys. Rev. 152, 1110–1114 (1966).
[CrossRef]

1953 (1)

K. Husimi, “Miscellanea in elementary quantum mechanics,” Prog. Theor. Phys. 9, 381–402 (1953); N. A. Chernikov, “System with time-dependent quadratic in x and p Hamiltonian,” Zh. Exp. Theor. Fiz. 53, 1006–1017 (1967).
[CrossRef]

1934 (1)

H. P. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46, 794–801 (1934).
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1930 (1)

E. Schrödinger, “Zum Heisenbergschen Unschärfeprinzip,” Sitzungsber. K. Preuss. Akad. Wiss. Phys. Math. Kl. 19, 296–303 (1930); H. P. Robertson, “A general formulation of the uncertainty principle and its classical interpretation,” Phys. Rev. 35, 667–667 (1930).

1929 (1)

H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
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1927 (2)

W. Heisenberg, “Uber den anschaulichen Inhalt der quantentheoretishen Kinematik and Mechanik,” Z. Phys. 43, 172–198 (1927).
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E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Z. Phys. 44, 326–352 (1927).
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1926 (1)

E. Schrödinger, “Der Stetige Übergang von den Mikro- zur Makromechanik,” Naturwissenschaften 14, 664–666 (1926).
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1880 (1)

V. Ermakov, “Second order differential equations. Integrability conditions in finite form,” Univ. Izv. 20(3), 1–25 (1880); H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1968).
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N. A. Ansari, V. I. Man’ko, “Photon statistics of multimode even and odd coherent light,” Phys. Rev. A 50, 1942–1945 (1994); V. V. Dodonov, V. I. Man’ko, D. E. Nikonov, “Even and odd coherent states for multimode parametric systems,” Phys. Rev. A 51, 3328–3336 (1995).
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C. Aragone, E. Chalbaud, S. Salamo, “On intelligent spin states,” J. Math. Phys. 17, 1963–1971 (1976).
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A. O. Barut, L. Girardello, “New ‘coherent’ states associated with noncompact groups,” Commun. Math. Phys. 21, 41–55 (1971).
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J. A. Bergou, M. Hillery, D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991); M. M. Nieto, D. R. Truax, “Squeezed states for general systems,” Phys. Rev. Lett. 71, 2843–2846 (1993).
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Bhamathi, G.

E. S. G. Sudarshan, C. B. Chiu, G. Bhamathi, “Generalized uncertainty relations and characteristic invariants for multimode states,” Phys. Rev. A 52, 43–54 (1995).
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G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999).
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S. L. Braunstein, C. M. Caves, G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. (N.Y.) 247, 135–175 (1996).
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Caves, C. M.

S. L. Braunstein, C. M. Caves, G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. (N.Y.) 247, 135–175 (1996).
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C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981); “Defense of the standard quantum limit for free-mass position,” Phys. Rev. Lett. 54, 2465–2468 (1985).
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C. Aragone, E. Chalbaud, S. Salamo, “On intelligent spin states,” J. Math. Phys. 17, 1963–1971 (1976).
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E. S. G. Sudarshan, C. B. Chiu, G. Bhamathi, “Generalized uncertainty relations and characteristic invariants for multimode states,” Phys. Rev. A 52, 43–54 (1995).
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Dodonov, V. V.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, A. Wünsche, “Energy-sensitive and ‘classical-like’ distances between quantum states,” Phys. Scr. 59, 81–89 (1999); D. A. Trifonov, S. G. Donev, “Polarized distances between quantum states and observables,” E-print quant-ph/0005087, http://arxiv.org/abs/quant-ph/yymmnnn .
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V. V. Dodonov, V. I. Man’ko, O. V. Man’ko, “Nonstationary quantum oscillator,” Proc. P. N. Lebedev Phys. Inst. 191, 171–244 (1990); A. K. Angelow, “Light propagation in nonlinear waveguide and classical two-dimensional oscillator,” Physica A 256, 485–498 (1998).
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V. V. Dodonov, V. I. Man’ko, “Generalizations of the uncertainty relations in quantum mechanics,” Proc. P. N. Lebedev Phys. Inst. 183, 3–70 (1987).

V. V. Dodonov, E. V. Kurmyshev, V. I. Man’ko, “Generalized uncertainty relation and correlated coherent states,” Phys. Lett. A 79, 150–152 (1980); V. V. Dodonov, V. I. Man’ko, “Invariants and correlated states of nonstationary quantum systems,” in Proc. P. N. Lebedev Phys. Inst. 183, 71–181 (1987).
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V. V. Dodonov, I. A. Malkin, V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator,” Physica (Amsterdam) 72, 597–618 (1974).
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V. V. Dodonov, V. I. Man’ko, “Universal invariants of quantum systems and generalized uncertainty relations,” in Group Theoretical Methods in Physics, M. A. Markov, V. I. Man’ko, A. E. Shabad, eds. (Harwood Academic, Chur, Switzerland, 1985), pp. 591–612.

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D. A. Trifonov, S. G. Donev, “Characteristic uncertainty relations,” J. Phys. A 31, 8041–8047 (1998).
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Eberly, J.

Ermakov, V.

V. Ermakov, “Second order differential equations. Integrability conditions in finite form,” Univ. Izv. 20(3), 1–25 (1880); H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1968).
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W.-M. Zhang, D. H. Feng, R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–924 (1990); S. Tareque Ali, J.-P. Antoine, J.-P. Gazeau, U. A. Mueler, “Coherent states and their generalizations: a mathematical overview,” Rev. Math. Phys. 7, 1013–1104 (1995).
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S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
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W.-M. Zhang, D. H. Feng, R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–924 (1990); S. Tareque Ali, J.-P. Antoine, J.-P. Gazeau, U. A. Mueler, “Coherent states and their generalizations: a mathematical overview,” Rev. Math. Phys. 7, 1013–1104 (1995).
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Girardello, L.

A. O. Barut, L. Girardello, “New ‘coherent’ states associated with noncompact groups,” Commun. Math. Phys. 21, 41–55 (1971).
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W. Heisenberg, “Uber den anschaulichen Inhalt der quantentheoretishen Kinematik and Mechanik,” Z. Phys. 43, 172–198 (1927).
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J. A. Bergou, M. Hillery, D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991); M. M. Nieto, D. R. Truax, “Squeezed states for general systems,” Phys. Rev. Lett. 71, 2843–2846 (1993).
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K. Husimi, “Miscellanea in elementary quantum mechanics,” Prog. Theor. Phys. 9, 381–402 (1953); N. A. Chernikov, “System with time-dependent quadratic in x and p Hamiltonian,” Zh. Exp. Theor. Fiz. 53, 1006–1017 (1967).
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Karlson, A.

G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999).
[CrossRef]

Katriel, J.

Kempf, A.

A. Kempf, G. Mangano, R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D 52, 1108–1118 (1995).
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Kennard, E. H.

E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Z. Phys. 44, 326–352 (1927).
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Klauder, J. R.

J. R. Klauder, B.-S. Skagerstam, Coherent States—Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

Knight, P.

R. Loudon, P. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
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V. V. Dodonov, E. V. Kurmyshev, V. I. Man’ko, “Generalized uncertainty relation and correlated coherent states,” Phys. Lett. A 79, 150–152 (1980); V. V. Dodonov, V. I. Man’ko, “Invariants and correlated states of nonstationary quantum systems,” in Proc. P. N. Lebedev Phys. Inst. 183, 71–181 (1987).
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R. Loudon, P. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
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A. Luis, J. Perina, “SU(2) coherent states in parametric down-conversion,” Phys. Rev. A 53, 1886–1893 (1996).
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V. V. Dodonov, I. A. Malkin, V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator,” Physica (Amsterdam) 72, 597–618 (1974).
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I. A. Malkin, V. I. Man’ko, “Coherent states and excitation of n-dimensional nonstationary forced oscillator,” Phys. Lett. A 32, 243–244 (1970); A. Holz, “N-dimensional anisotropic oscillator in a time-dependent homogeneous electromagnetic field,” Lett. N. Cimento A 4, 1319–1323 (1970); I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Dynamical symmetry of nonstationary systems,” Nuovo Cimento A 4, 773–793 (1971).
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I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Invariants and evolution of coherent states of charged particles in a time dependent magnetic field,” Phys. Lett. A 30, 414–414 (1969); “Coherent states and transition probabilities in a time dependent electromagnetic field,” Phys. Rev. D 2, 1371–1385 (1970).
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I. A. Malkin, V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems (Nauka, Moscow, 1979).

Man’ko, O. V.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, A. Wünsche, “Energy-sensitive and ‘classical-like’ distances between quantum states,” Phys. Scr. 59, 81–89 (1999); D. A. Trifonov, S. G. Donev, “Polarized distances between quantum states and observables,” E-print quant-ph/0005087, http://arxiv.org/abs/quant-ph/yymmnnn .
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V. V. Dodonov, V. I. Man’ko, O. V. Man’ko, “Nonstationary quantum oscillator,” Proc. P. N. Lebedev Phys. Inst. 191, 171–244 (1990); A. K. Angelow, “Light propagation in nonlinear waveguide and classical two-dimensional oscillator,” Physica A 256, 485–498 (1998).
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Man’ko, V. I.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, A. Wünsche, “Energy-sensitive and ‘classical-like’ distances between quantum states,” Phys. Scr. 59, 81–89 (1999); D. A. Trifonov, S. G. Donev, “Polarized distances between quantum states and observables,” E-print quant-ph/0005087, http://arxiv.org/abs/quant-ph/yymmnnn .
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V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
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N. A. Ansari, V. I. Man’ko, “Photon statistics of multimode even and odd coherent light,” Phys. Rev. A 50, 1942–1945 (1994); V. V. Dodonov, V. I. Man’ko, D. E. Nikonov, “Even and odd coherent states for multimode parametric systems,” Phys. Rev. A 51, 3328–3336 (1995).
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V. V. Dodonov, V. I. Man’ko, O. V. Man’ko, “Nonstationary quantum oscillator,” Proc. P. N. Lebedev Phys. Inst. 191, 171–244 (1990); A. K. Angelow, “Light propagation in nonlinear waveguide and classical two-dimensional oscillator,” Physica A 256, 485–498 (1998).
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V. V. Dodonov, V. I. Man’ko, “Generalizations of the uncertainty relations in quantum mechanics,” Proc. P. N. Lebedev Phys. Inst. 183, 3–70 (1987).

V. V. Dodonov, E. V. Kurmyshev, V. I. Man’ko, “Generalized uncertainty relation and correlated coherent states,” Phys. Lett. A 79, 150–152 (1980); V. V. Dodonov, V. I. Man’ko, “Invariants and correlated states of nonstationary quantum systems,” in Proc. P. N. Lebedev Phys. Inst. 183, 71–181 (1987).
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V. V. Dodonov, I. A. Malkin, V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator,” Physica (Amsterdam) 72, 597–618 (1974).
[CrossRef]

I. A. Malkin, V. I. Man’ko, “Coherent states and excitation of n-dimensional nonstationary forced oscillator,” Phys. Lett. A 32, 243–244 (1970); A. Holz, “N-dimensional anisotropic oscillator in a time-dependent homogeneous electromagnetic field,” Lett. N. Cimento A 4, 1319–1323 (1970); I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Dynamical symmetry of nonstationary systems,” Nuovo Cimento A 4, 773–793 (1971).
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I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Invariants and evolution of coherent states of charged particles in a time dependent magnetic field,” Phys. Lett. A 30, 414–414 (1969); “Coherent states and transition probabilities in a time dependent electromagnetic field,” Phys. Rev. D 2, 1371–1385 (1970).
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I. A. Malkin, V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems (Nauka, Moscow, 1979).

V. I. Man’ko, “Coherent state method for arbitrary dynamical systems,” in Novosti Fundamentalnoy Fiziki, V. I. Man’ko, ed. (Mir, Moscow, 1972), Vol. 1, pp. 5–25.

V. V. Dodonov, V. I. Man’ko, “Universal invariants of quantum systems and generalized uncertainty relations,” in Group Theoretical Methods in Physics, M. A. Markov, V. I. Man’ko, A. E. Shabad, eds. (Harwood Academic, Chur, Switzerland, 1985), pp. 591–612.

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S. Mancini, “Even and odd nonlinear coherent states,” Phys. Lett. A 233, 291–296 (1997); S. Sivakumar, “Generation of even and odd nonlinear coherent states,” E-print quant-ph/9902054, http://arxiv.org/abs/quant-ph/yymmnnn ; B. Roy, P. Roy, “Phase properties of even and odd nonlinear coherent states,” Phys. Lett. A 257, 264–268 (1999); “Time dependent nonclassical properties of even/odd nonlinear coherent states,” Phys. Lett. A 263, 48–52 (1999).
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A. Kempf, G. Mangano, R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D 52, 1108–1118 (1995).
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C. Brif, A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A 54, 4505–4518 (1996).
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Mann, R. B.

A. Kempf, G. Mangano, R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D 52, 1108–1118 (1995).
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V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
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McDermott, R. J.

R. J. McDermott, A. I. Solomon, “Squeezed states parametrized by elements of noncommutative algebras,” Czech. J. Phys. 46, 235–241 (1996).
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S. L. Braunstein, C. M. Caves, G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. (N.Y.) 247, 135–175 (1996).
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M. M. Miller, E. A. Mishkin, “Characteristic states of the electromagnetic radiation field,” Phys. Rev. 152, 1110–1114 (1966).
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R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 37, 3868–3880 (1987); X. Ma, W. Rhodes, “Multimode squeeze operators and squeezed states,” Phys. Rev. A 41, 4624–4631 (1990); V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, “Multidimensional Hermite polynomials and photon distribution for polymode mixed light,” Phys. Rev. A 50, 813–817 (1994).
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H. P. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46, 794–801 (1934).
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H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
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C. Aragone, E. Chalbaud, S. Salamo, “On intelligent spin states,” J. Math. Phys. 17, 1963–1971 (1976).
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E. Schrödinger, “Zum Heisenbergschen Unschärfeprinzip,” Sitzungsber. K. Preuss. Akad. Wiss. Phys. Math. Kl. 19, 296–303 (1930); H. P. Robertson, “A general formulation of the uncertainty principle and its classical interpretation,” Phys. Rev. 35, 667–667 (1930).

E. Schrödinger, “Der Stetige Übergang von den Mikro- zur Makromechanik,” Naturwissenschaften 14, 664–666 (1926).
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R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 37, 3868–3880 (1987); X. Ma, W. Rhodes, “Multimode squeeze operators and squeezed states,” Phys. Rev. A 41, 4624–4631 (1990); V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, “Multidimensional Hermite polynomials and photon distribution for polymode mixed light,” Phys. Rev. A 50, 813–817 (1994).
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Skagerstam, B.-S.

J. R. Klauder, B.-S. Skagerstam, Coherent States—Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

Söderholm, J.

G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999).
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Solomon, A. I.

R. J. McDermott, A. I. Solomon, “Squeezed states parametrized by elements of noncommutative algebras,” Czech. J. Phys. 46, 235–241 (1996).
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J. Katriel, A. I. Solomon, G. D’Ariano, M. Rasetti, “Multiphoton squeezed states,” J. Opt. Soc. Am. B 4, 1728–1735 (1987).
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V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
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R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 37, 3868–3880 (1987); X. Ma, W. Rhodes, “Multimode squeeze operators and squeezed states,” Phys. Rev. A 41, 4624–4631 (1990); V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, “Multidimensional Hermite polynomials and photon distribution for polymode mixed light,” Phys. Rev. A 50, 813–817 (1994).
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Sudarshan, E. S. G.

E. S. G. Sudarshan, C. B. Chiu, G. Bhamathi, “Generalized uncertainty relations and characteristic invariants for multimode states,” Phys. Rev. A 52, 43–54 (1995).
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Trifonov, A.

G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999).
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Trifonov, D. A.

D. A. Trifonov, “State extended uncertainty relations,” J. Phys. A 33, L299–L304 (2000), E-print quant-ph/0005086, http://arxiv.org/abs/quant-ph/yymmnnn .
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D. A. Trifonov, “On the squeezed states for n observables,” Phys. Scr. 58, 246–255 (1998).
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D. A. Trifonov, S. G. Donev, “Characteristic uncertainty relations,” J. Phys. A 31, 8041–8047 (1998).
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D. A. Trifonov, “Barut–Girardello coherent states for u(p, q) and sp(N, R) and their macroscopic superpositions,” J. Phys. A 31, 5673–5696 (1998).
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D. A. Trifonov, “On the stable evolution of squeezed and correlated states,” J. Sov. Laser Res. 12, 414–420 (1991); “Completeness and geometry of Schrödinger minimum uncertainty states,” J. Math. Phys. 34, 100–110 (1993).
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D. A. Trifonov, “On coherent states of quantum systems and uncertainty relations,” Bulg. J. Phys. 2, 303–311 (1975).

I. A. Malkin, V. I. Man’ko, D. A. Trifonov, “Invariants and evolution of coherent states of charged particles in a time dependent magnetic field,” Phys. Lett. A 30, 414–414 (1969); “Coherent states and transition probabilities in a time dependent electromagnetic field,” Phys. Rev. D 2, 1371–1385 (1970).
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D. A. Trifonov, “Uncertainty matrix, multimode squeezed states and generalized even and odd coherent states,” Preprint INRNE-TH-95/5 (1995); available on request from D. A. Trifonov.

D. A. Trifonov, “The uncertainty way of generalizations of coherent states,” in Geometry, Integrability and Quantization, I. M. Mladenov, G. L. Naber, eds. (Coral, Sofia, Bulgaria, 2000), pp. 257–282 [quant-ph/9912084], http://arxiv.org/abs/quant-ph/yymmnnn . Note that in Eq. (5) the factor 2ℏ/m should be replaced by (2ℏ/mω0)1/2.

Tsegaye, T.

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V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, A. Wünsche, “Energy-sensitive and ‘classical-like’ distances between quantum states,” Phys. Scr. 59, 81–89 (1999); D. A. Trifonov, S. G. Donev, “Polarized distances between quantum states and observables,” E-print quant-ph/0005087, http://arxiv.org/abs/quant-ph/yymmnnn .
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D. A. Trifonov, “Uncertainty matrix, multimode squeezed states and generalized even and odd coherent states,” Preprint INRNE-TH-95/5 (1995); available on request from D. A. Trifonov.

D. A. Trifonov, “Algebraic coherent states and squeezing,” E-print quant-ph/9609001, http://arxiv.org/abs/quant-ph/yymmnnn ; “Schrödinger intelligent states and linear and quadratic amplitude squeezing,” E-print quant-ph/9609017, http://arxiv.org/abs/quant-ph/yymmnnn .

B. Nagel, “Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics,” in Modern Group Theoretical Methods in Physics, J. Bertrand, M. Flator, J.-P. Gazeau, M. Irac-Astaud, D. Sternheimer, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 211–220.

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V. V. Dodonov, V. I. Man’ko, “Universal invariants of quantum systems and generalized uncertainty relations,” in Group Theoretical Methods in Physics, M. A. Markov, V. I. Man’ko, A. E. Shabad, eds. (Harwood Academic, Chur, Switzerland, 1985), pp. 591–612.

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J. Uffink, “Two new kinds of uncertainty relations,” in Proceedings of the Third International Workshop on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, N. H. Rubin, Y. Shih, W. W. Zachary, eds., NASA Conf. Publ.3720, 155–160 (1993); S. Kudaka, S. Matsumoto, “Uncertainty principle for proper time and mass,” J. Math. Phys. 40, 1237–1245 (1999).
[CrossRef]

D. A. Trifonov, “The uncertainty way of generalizations of coherent states,” in Geometry, Integrability and Quantization, I. M. Mladenov, G. L. Naber, eds. (Coral, Sofia, Bulgaria, 2000), pp. 257–282 [quant-ph/9912084], http://arxiv.org/abs/quant-ph/yymmnnn . Note that in Eq. (5) the factor 2ℏ/m should be replaced by (2ℏ/mω0)1/2.

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Equations (38)

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(ΔX)2(ΔY)2¼|[X,Y]|2.
(Δq)2+(Δp)21,
1=|αα|dμ(α),dμ(α)=1πd2α.
A(t)|α; t=α|α; t,
A(t)=u(t)a+v(t)a=A(u, v).
Ψα(x, t)=x|α; t=(πl02)-1/4(u-v)1/2exp-12l02v+uu-vx-2l0αu+v2+12u*+v*u+vα2-|α|2,
K3=aa/2+1/4,K-=a2/2,K+=a2/2.
|α, u(t), v(t)=U(t)|α=U(t)D(α)|0.
Aμ(u, v)|α, u, v=αμ|α, u, v,μ=1,, s.
|a, u, v=exp(i arg u)exp(ζK+-ζ*K-)|α.
|ξ; k=exp(ζK+-ζ*K-)|k, k=(1-|ξ|2)k exp(ξK+)|k, k,
|z; k=NBGn=0zn[n!Γ(2k+n)]1/2|k, k+n,
(ΔX)2+(ΔY)2|[X, Y]|
(Δp)2(Δq)2-(Δpq)21/4,
dpdt=-Hq,dqdt=Hp,
dp˜dt=-Hq˜,dq˜dt=Hp˜.
(ΔX)2(ΔY)21/4|[X, Y]|2+(ΔXY)2,
[u(X-iY)+v(X+iY)]|z, u, v=z|z, u, v,
(ΔX)2=12|u-v|2|u|2-|v|2i[X, Y],
ΔXY=Im(u*v)|u|2-|v|2i[X, Y],
(ΔY)2=12|u+v|2|u|2-|v|2i[X, Y].
¼[(ΔX)2+(ΔY)2]2
(ΔX)2(ΔY)2(ΔX)2(ΔY)2-(ΔXY)2¼|[X, Y]|2,
{|z}{|z, u, v|Im uv*=0}{|z, u, v},
det σ(X)det C(X),
A˜μ(u, v)uμνa˜ν+vμνa˜ν=βμjXj,
A˜μ(u, v)|z, u, v=zμ|z, u, v,
σ=B-10C˜C˜T0B-1T,B=u+vi(u-v)u*+v*i(v*-u*),
|z, u, v, w; k=Nn=0-l+w2un(2κ)nn!1/2× 2F1κ+zl, -n; 2κ;2ll+w|k, n+k,
(βiKi+βjKj)|ψ=zij|ψ,i<j,
0=det(M-λ)=r=0nCr(n)(M)(-λ)n-r.
Cr(n)[σ(X)]Cr(n)[C(X)],
Cr(n)mσ(X; ρm)Cr(n)mC(X; ρm).
detmσ(X, ρm)detmC(X, ρm).
½[ΔXX(ψ1)ΔYY(ψ2)+ΔXX(ψ2)ΔYY(ψ1)]-ΔXY(ψ1)ΔXY(ψ2)1/4ψ1|[X, Y]|ψ1ψ2|[Y, X]|ψ2,
{[ΔX(ψ1)]2+1|X|12}{[(ΔX(ψ2)]2+2|X|22}|1|X2|2|2.
g(ψ1; ψ2; X)=|ψ2|X2|ψ1|(ψ1|X2|ψ1ψ2|X2|ψ2)1/2
Pr2(X, ρ)+Vr2(X, ρ)1.

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