OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2486–2495

Generalized uncertainty relations and coherent and squeezed states

D. A. Trifonov  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2486-2495 (2000)

View Full Text Article

Enhanced HTML    Acrobat PDF (224 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.6570) Quantum optics : Squeezed states

Original Manuscript: March 9, 2000
Revised Manuscript: July 20, 2000
Manuscript Accepted: July 31, 2000
Published: December 1, 2000

D. A. Trifonov, "Generalized uncertainty relations and coherent and squeezed states," J. Opt. Soc. Am. A 17, 2486-2495 (2000)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. W. Heisenberg, “Uber den anschaulichen Inhalt der quantentheoretishen Kinematik and Mechanik,” Z. Phys. 43, 172–198 (1927). [CrossRef]
  2. E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Z. Phys. 44, 326–352 (1927). [CrossRef]
  3. H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929). [CrossRef]
  4. M. Ozawa, “Quantum limits of measurements and uncertainty principle,” in Quantum Aspects of Optical Communications, C. Bendjaballak, O. Hirota, S. Reynaud, eds., Vol. 378 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1991), pp. 3–17.
  5. 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]
  6. J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detection,” Phys. Rev. D 19, 1669–1679 (1979). [CrossRef]
  7. H. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976). [CrossRef]
  8. J. R. Klauder, B.-S. Skagerstam, Coherent States—Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).
  9. 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]
  10. 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]
  11. 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).
  12. 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]
  13. K. Wodkiewicz, J. Eberly, “Coherent states, squeezed fluctuations and the SU(2) and SU(1, 1) groups in quantum optics applications,” J. Opt. Soc. Am. B 2, 458–466 (1985). [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. 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.
  19. 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 .
  20. C. Brif, “Two-photon algebra eigenstates. A unified approach to squeezing,” Ann. Phys. (N.Y.) 251, 180–207 (1996). [CrossRef]
  21. 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]
  22. D. A. Trifonov, “Robertson intelligent states,” J. Phys. A 30, 5941–5957 (1997). [CrossRef]
  23. D. A. Trifonov, “On the squeezed states for n observables,” Phys. Scr. 58, 246–255 (1998). [CrossRef]
  24. D. A. Trifonov, S. G. Donev, “Characteristic uncertainty relations,” J. Phys. A 31, 8041–8047 (1998). [CrossRef]
  25. G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, A. Karlson, “Complementarity and uncertainty relations,” Phys. Rev. A 60, 1874–1882 (1999). [CrossRef]
  26. M. M. Miller, E. A. Mishkin, “Characteristic states of the electromagnetic radiation field,” Phys. Rev. 152, 1110–1114 (1966). [CrossRef]
  27. 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]
  28. I. A. Malkin, V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems (Nauka, Moscow, 1979).
  29. 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]
  30. 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]
  31. E. Schrödinger, “Der Stetige Übergang von den Mikro- zur Makromechanik,” Naturwissenschaften 14, 664–666 (1926). [CrossRef]
  32. 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.
  33. 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.
  34. D. A. Trifonov, “On coherent states of quantum systems and uncertainty relations,” Bulg. J. Phys. 2, 303–311 (1975).
  35. J. Katriel, A. I. Solomon, G. D’Ariano, M. Rasetti, “Multiphoton squeezed states,” J. Opt. Soc. Am. B 4, 1728–1735 (1987). [CrossRef]
  36. 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). [CrossRef]
  37. 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]
  38. H. P. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46, 794–801 (1934). [CrossRef]
  39. D. A. Stoler, “Equivalent classes of minimum uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970). [CrossRef]
  40. R. Loudon, P. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987). [CrossRef]
  41. 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]
  42. 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]
  43. A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys. 26, 222–236 (1972). [CrossRef]
  44. A. O. Barut, L. Girardello, “New ‘coherent’ states associated with noncompact groups,” Commun. Math. Phys. 21, 41–55 (1971). [CrossRef]
  45. K. Fujii, K. Funahashi, “Extension of the Barut–Girardello coherent state and path integral,” J. Math. Phys. 38, 4422–4434 (1997). [CrossRef]
  46. 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]
  47. 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.
  48. 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]
  49. 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]
  50. C. Aragone, E. Chalbaud, S. Salamo, “On intelligent spin states,” J. Math. Phys. 17, 1963–1971 (1976). [CrossRef]
  51. C. Brif, A. Mann, “Nonclassical interferometry with intelligent light,” Phys. Rev. A 54, 4505–4518 (1996). [CrossRef] [PubMed]
  52. A. Luis, J. Perina, “SU(2) coherent states in parametric down-conversion,” Phys. Rev. A 53, 1886–1893 (1996). [CrossRef] [PubMed]
  53. 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]
  54. 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]
  55. R. J. McDermott, A. I. Solomon, “Squeezed states parametrized by elements of noncommutative algebras,” Czech. J. Phys. 46, 235–241 (1996). [CrossRef]
  56. 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]
  57. D. A. Trifonov, “Exact solution for the general nonstationary oscillator with a singular perturbation,” J. Phys. A 32, 3649–3661 (1999). [CrossRef]
  58. F. R. Gantmaher, Teoria Matrits (Nauka, Moscow, 1975).
  59. 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]
  60. 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]
  61. 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).
  62. 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.
  63. 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]
  64. A. Kempf, G. Mangano, R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D 52, 1108–1118 (1995). [CrossRef]
  65. 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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited