Russian Microelectronics

, Volume 35, Issue 1, pp 7–20 | Cite as

Schmidt modes and entanglement in continuous-variable quantum systems

  • A. Yu. Bogdanov
  • Yu. I. Bogdanov
  • K. A. Valiev


The extraction of Schmidt modes for continuous-variable systems is considered. An algorithm based on the singular-value decomposition of a matrix is proposed. It is applied to the entanglement in (i) an atom—photon system with spontaneous emission and (ii) a system of biphotons with spontaneous parametric downconversion (SPDC) of type II. For the atom—photon system, the evolution of entangled states is found to be governed by a parameter approximately equal to the fine-structure constant times the atom-to-electron mass ratio. An analysis is made of the dynamics of atom—photon entanglement on the assumption that the system’s evolution is determined by the superposition of an initial and a final state. It is shown that in the course of emission the entanglement entropy first rises on a timescale of order the excited-state lifetime and then falls, approaching asymptotically a residual level due to the initial energy spread of the atomic packet (momentum spread squared). SPDC of type II is analyzed by means of the polarization density matrix and a newly introduced coherence parameter for two spatially separated modes. The loss of intermodal coherence is addressed that results from the difference in behavior between ordinary-and extraordinary-ray photons in a nonlinear crystal. The degree of intermodal coherence is investigated as a function of the product of crystal length and pump bandwidth.


Entangle State Entanglement Entropy Nonlinear Crystal Crystal Length Polarization Density 
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Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • A. Yu. Bogdanov
    • 2
  • Yu. I. Bogdanov
    • 1
  • K. A. Valiev
    • 1
  1. 1.Institute of Physics and TechnologyRussian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia

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