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Quantum Fluctuations in Unstable Optical Systems


Quantum Fluctuations in Unstable Optical Systems

Tech Area / Field

  • PHY-OPL/Optics and Lasers/Physics

3 Approved without Funding

Registration date

Leading Institute
Institute for Physical Research, Armenia, Ashtarak-2


  • University of Arkansas / Fulbright College of Arts and Sciences, USA, AR, Fayetteville

Project summary

For some nonlinear optical systems (for example, with intracavity generation of second and third harmonics [1-3] or four-wave mixing [4]) steady-state solutions exist only for relatively low intensities of pumping fields. Above some critical strengths of the driving field the dynamics of the photon numbers in the modes, with a semiclassical approach, become unstable and move to the self-pulsing regime [1]. Second harmonic generation (SHG) in a two-mode cavity with a strong coherent input to the fundamental mode is one of the most simple and convenient systems which can be used to investigate the basic aspects of optical instability and therefore it is often used for demonstrating the efficiency of new numerical methods in quantum optics [5-12]. The steady-state solution for this system has been obtained exactly [2]. Doertle and Schenzle [5] have analysed SHG numerically by solving the Langevin stochastic equations, which are equivalent to the Fokker-Planck equations for the positive P-representation (PPR) [13]. Furthermore, SHG was studied in the semiclassical limit using the dynamical equation for the Wigner function [6], an equation obtained with the help of Q-distribution [7], and also using the method of cumulants [8]. More recently, in studies of SHG the quantum state diffusion (QSD) technique was employed [9,10] and subsequently the QSD method with a moving basis was used [11]. Both of these methods are based on numerical solution of the quantum stochastic Schroedinger equation [14]. Thereafter, Zheng and Savage [12] solved the SHG problem in the chaotic regime by applying the quantum jump technique [15,16] and calculating the density matrix directly. We presented above the analysis of works concerning the theory of unstable optical systems. Note that there is no quantum theory describing the dynamics of four-wave mixing. The goal of the present Project is investigation of quantum fluctuations around the bifurcation point of an optical system in the four-wave mixing process. Successful solution of this problem will give additional new knowledge in the field of the quantum theory of unstable optical systems. It is apparent from the above analysis of literature that the quantum theory of unstable optical systems looks like solution of separate problems. We intend, after successful implementation of the work on the theory of unstable behavior of the four-wave mixing, to work on generalization of quantum theories of unstable optical systems. This is important for the further development of the theory of unstable optical phenomena and will be useful for the design of experiments in the proposed field of research. We expect a strong dependence of the behavior of physical quantities on the distance of the optical system from the bifurcation point. Also a strong dependence of these quantities on the initial state of the optical system is expected. We expect as well an increase of quantum fluctuations when the optical system moves from the bifurcation point to the right. In this case the dispersion of the number of photons and the phase of the optical system modes must grow. Depending on the initial coherent states of interacting fields the distribution functions of optical system mode phases may, as expected, be in the oscillatory regime. By analogy with the processes of intracavity SHG and THG we guess that the modes in four-wave mixing will be, in the instability region, in the two-component phase state. It is expected that the quantum entropy of the modes of the optical system will increase when the system moves from the bifurcation point to the right.

The Project we propose will add new knowledge in understanding the behavior of unstable optical systems.The working team is competent in this field of research and is capable of solving the given problem. The manager of the proposed Project, S.T.Gevorgyan has an experience in research into the quantum dynamics of unstable optical systems. His is author or co-author of many works concerning this field of research, in particular, [3,17-19]. We expect that the results of studies of the problem proposed will be interesting for the groups of physicists active in the theory or experiments dealing with unstable optical systems. We guess that the results of our studies will assist the designing of new experiments and may be used for further development of the theory of unstable optical phenomena. The Project corresponds to the goals of ISTC. It gives a possibility for scientists active in the developing of armament to concentrate their effort in the peaceful activity. The Project also promotes the integration of scientists from Armenia into the international scientific community. The duration of the Project is 12 months. The Project consists of one problem that will be solved by two persons: a defense specialist and the manager. There will be, during the overall duration of the Project, an exchange of information and scientific concepts with the foreign collaborator, as well as joint effort for the solution of the problem and publication of joint articles. For solving the proposed problems we intend to employ two techniques of simulation. The first will be as follows. We will obtain in PPR the equation of motion for the P-function of the optical system [20]. With use of the Ito rule we will obtain from the Fokker-Planck equation the system of Langevin equations for the stochastic field amplitudes [20]. From the ensemble of realizations of stochastic amplitudes of the fields we will obtain the distribution function for physical quantities of the system [17]. The second technique will be that of quantum jumps [21]. We will compute the matrix of quantum trajectories of modes of the optical system. The density matrices of system modes we will determine as the mathematical expectation of an ensemble of trajectory matrices. The quantum entropy of the optical system modes will be calculated by means of numerical diagonalization of the mode density matrices. The Wigner functions of the optical system modes will be calculated following the corresponding formulas of the work [22].


  1. P. D. Drummond, K. J. McNeil, and D. F. Walss, Optica Acta, 27, 321 (1980).
  2. P. D. Drummond, K. J. McNeil, and D. F. Walls, Optica Acta, 28, 211 (1981).
  3. S. T. Gevorkyan, G. Yu. Kryuchkyan, K. V. Kheruntsyan, Optics Comm., 134, 440 (1997).
  4. L. A. Lugiato, Progr. in Optics, XXI, 68 (1984)
  5. M. Dorfle, and A. Schenzle, Z. Phys. B, 65, 113 (1986)
  6. M. Dorfle, and R. Graham, Optical instabilities, edited by R. U. Boyd, M. G. Raymer and L. M. Narducci (Cambridge; Cambridge University Press) 352 (1986)
  7. C. M. Savage, Phys. Rev. A, 37, 158 (1988).
  8. R. Schack, and A. Schenzle, Phys. Rev. A, 41, 3847 (1990).
  9. P. Goetsch, and R. Graham, Ann. Phys. Lpz, 2, 706 (1993).
  10. N. Gisin, and I. C. Percival, J. Phys. A, 25, 5677
  11. R. Schack, T. A. Brun, and I. C. Percival, J. Phys. A 28, 1995 (1995).
  12. X. P. Zheng, and C. M. Savage, Phys. Rev. A, 51, 792 (1995).
  13. P. D. Drummond, and C. W. Gardiner, J. Phys. A, 13, 2353 (1980).
  14. N. Gisin, Phys. Rev. Lett., 52, 1657 (1984).
  15. H. J. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics (Berlin: Springer-Verlag) (1994).
  16. W. L. Power, and P. L. Knight, Phys. Rev. A, 53, 1052 (1996).
  17. S. T. Gevorkyan, Phys. Rev. A, 62, 013813 (2000).
  18. S. T. Gevorkyan, G. Yu. Kryuchkyan, and N. T. Muradyan, Phys. Rev. A, 61, 043805 (2000).
  19. S. T. Gevorkyan, Phys. Rev. A, 58, 4862 (1998).
  20. C. W. Gardiner, Handbook of Stochastic Methods, 2 nd ed. (Berlin: Springer) (1985).
  21. K. Molmer, Y. Gastin, and J. Dalibard, J. Opt. Soc. Am. B, 10, 524 (1992).
  22. L. Gilles, B. M. Garraway, and P. L. Knight, Phys. Rev. A, 49, 2785 (1995)


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