Foundations of Physics Letters

, Volume 17, Issue 6, pp 507–534 | Cite as

Entanglement in the Interaction Between Two Quantum Oscillators Systems


No Heading

The fundamental quantum dynamics of two interacting oscillator systems are studied in two different scenarios. In one case, both oscillators are assumed to be linear, whereas in the second case, one oscillator is linear and the other is a non-linear, angular-momentum oscillator; the second case is, of course, more complex in terms of energy transfer and dynamics. These two scenarios have been the subject of much interest over the years, especially in developing an understanding of modern concepts in quantum optics and quantum electronics. In this work, however, these two scenarios are utilized to consider and discuss the salient features of quantum behaviors resulting from the interactive nature of the two oscillators, i.e., coherence, entanglement, spontaneous emission, etc., and to apply a measure of entanglement in analyzing the nature of the interacting systems.

The Heisenberg equation for both coupled oscillator scenarios are developed in terms of the relevant reduced kinematics operator variables and parameterized commutator relations. For the second scenario, by setting the relevant commutator relations to one or zero, respectively, the Heisenberg equations are able to describe the full quantum or classical motion of the interaction system, thus allowing us to discern the differences between the fully quantum and fully classical dynamical picture.

For the coupled linear and angular-momentum oscillator system in the fully quantum-mechanical description, we consider special examples of two, three, four-level angular momentum systems, demonstrating the explicit appearances of entanglement. We also show that this entanglement persists even as the coupled angular momentum oscillator is taken to the limit of a large number of levels, a limit which would go over to the classical picture for an uncoupled angular momentum oscillator.

Key words:

entanglement coupled-boson representation spontaneous emission 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    1. P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE, Los Alamitos, CA, 1994), pp. 124–134.Google Scholar
  2. 2.
    2. L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (ACM Press, New York, 1996), pp. 212–219.Google Scholar
  3. 3.
    3. I. R. Senitzky, “Nonperturbative analysis of the resonant interaction between a linear and a nonlinear oscillator,” Phys. Rev. A 3, 421 (1971).ADSGoogle Scholar
  4. 4.
    4. J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, B.-G. Englert, ed. (Springer, Berlin, 2001).MATHGoogle Scholar
  5. 5.
    5. A. Shalom and J. Zak, “A quantum mechanical model of interference,” Phys. Lett. A 43, 13 (1973).ADSGoogle Scholar
  6. 6.
    6. G. J. Iafrate and M. Croft, “Interaction between two coupled oscillators,” Phys. Rev. A 12, 1525 (1975).CrossRefADSGoogle Scholar
  7. 7.
    7. For a detailed discussion on general entanglement measures, see, e.g., O. Rudolph, “A new class of entanglement measures,” J. Math. Phys. 42, 5306 (2001).MATHADSMathSciNetGoogle Scholar
  8. 8.
    8. See, e.g., D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, 1996).MATHGoogle Scholar
  9. 9.
    9. G. Mahler and V. A. Weberruss, Quantum Networks: Dynamics of Open Nanostructures, 2nd edn. (Springer, New York, 1998).Google Scholar
  10. 10.
    10. L. P. Hughston, R. Jozsa, and W. K. Wootters, “A complete classification of quantum ensembles having a given density matrix”, Phys. Lett. A 183, 14 (1993). In an appendix therein, it was shown that for pure states |ψ〉 of the coupled system the two reduced density matrices ^ρ(1) and ^ρ(2), have the same non-zero eigenvalues with the same multiplicities. From this we easily find that Ρ[^ρ(1)] = Ρ[^ρ(2)].ADSMathSciNetGoogle Scholar
  11. 11.
    11. See, e.g., Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (National Bureau of Standards, 1964).Google Scholar
  12. 12.
    12. See, e.g., W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).Google Scholar
  13. 13.
    13. I. R. Senitzky, “Quantum optics. III. Two-frequency interactions,” Phys. Rev. 183, 1069 (1969).ADSGoogle Scholar
  14. 14.
    14. I. R. Senitzky, “Comment on energy balance for a dissipative system,” Phys. Rev. E 51, 5166 (1995).ADSGoogle Scholar
  15. 15.
    15. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” IEEE 51, 89 (1963).CrossRefGoogle Scholar
  16. 16.
    16. In Ref. [12] pp. 212, the diagonalization of Eq. (52) for j ( ) only was described in the same basis.Google Scholar
  17. 17.
    17. E. Abate and H. Haken, “Exakte Behandlung eines Laser-Modells,” Z. Naturforsh. 19a, 857 (1964)Google Scholar
  18. 17.
    M. Tavis and F. W. Cummings, “Exact solution for an N-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379 (1968).ADSGoogle Scholar
  19. 17.
    D. F. Walls and R. Barakat, “Quantum-mechanical amplification and frequency conversion with a trilinear Hamiltonian,” Phys. Rev. A 1, 446 (1970).ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc 2004

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringNorth Carolina State UniversityRaleigh

Personalised recommendations