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10 - Quantum field dynamics

Published online by Cambridge University Press:  05 May 2014

Peter D. Drummond
Affiliation:
Swinburne University of Technology, Victoria
Mark Hillery
Affiliation:
Hunter College, City University of New York
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Summary

In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful.

Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ(3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features.

This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction.

The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.

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Publisher: Cambridge University Press
Print publication year: 2014

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References

S. J., Carter and P. D., Drummond, J. Opt. Soc. Am. B 4, 1565 (1987).
S. J., Carter and P. D., Drummond, Phys. Rev. Lett. 67, 3757 (1991).
Y., Lai and H. A., Haus, Phys. Rev. A 40, 844 (1989).
E. H., Lieb and W., Liniger, Phys. Rev. 130, 1605 (1963).
C. N., Yang, Phys. Rev. 130, 1920 (1967).
P. D., Drummond and M. G., Raymer, Phys. Rev. A 44, 2072 (1991).
S. L., McCall and E. L., Hahn, Phys. Rev. A 83, 457 (1969).
Z. Y., Ou, L. J., Wang, and L., Mandel, Phys. Rev. A 40, 1428 (1989).

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  • Quantum field dynamics
  • Peter D. Drummond, Swinburne University of Technology, Victoria, Mark Hillery, Hunter College, City University of New York
  • Book: The Quantum Theory of Nonlinear Optics
  • Online publication: 05 May 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9780511783616.012
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  • Quantum field dynamics
  • Peter D. Drummond, Swinburne University of Technology, Victoria, Mark Hillery, Hunter College, City University of New York
  • Book: The Quantum Theory of Nonlinear Optics
  • Online publication: 05 May 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9780511783616.012
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Quantum field dynamics
  • Peter D. Drummond, Swinburne University of Technology, Victoria, Mark Hillery, Hunter College, City University of New York
  • Book: The Quantum Theory of Nonlinear Optics
  • Online publication: 05 May 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9780511783616.012
Available formats
×