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Journal of Contemporary Physics (Armenian Academy of Sciences)

Quantum Dynamics of Non-Degenerate Parameter Decay in an Optical Cavity

S. T. Gevorgyan, M. S. Gevorgyan

Published Dec 1, 2022• 2 citations
Publisher

Abstract

The quantum dynamics of the number of photons, of the quantum entropy, and the Wigner functions of field interacting modes are studied for the process of non-degenerate parametric decay in an optical cavity, where a quantum with energy $$\hbar {{\omega }_{3}}$$ decays into two photons with energies $$\hbar {{\omega }_{2}}$$ and $$\hbar {{\omega }_{1}}$$ ( $$\hbar {{\omega }_{3}} = \hbar {{\omega }_{2}} + \hbar {{\omega }_{1}}$$ ). It is shown that the mode with frequency $${{\omega }_{3}}$$ can localize in a squeezed state in the region of very strong coupling between the interacting modes, and the modes with frequencies $${{\omega }_{1}}$$ and $${{\omega }_{2}}$$ localize in states that are statistical mixtures of Fock states regardless of the value of the coupling coefficient

DOI
10.1134/s106833722204017x

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The quantum dynamics of the number of photons, of the quantum entropy, and the Wigner functions of field interacting modes are studied for the process of non-degenerate parametric decay in an optical cavity, where a quantum with energy ℏω3\hbar {{\omega }_{3}}ℏω3​ decays into two photons with energies ℏω2\hbar {{\omega }_{2}}ℏω2​ and ℏω1\hbar {{\omega }_{1}}ℏω1​ ( ℏω3=ℏω2+ℏω1\hbar {{\omega }_{3}} = \hbar {{\omega }_{2}} + \hbar {{\omega }_{1}}ℏω3​=ℏω2​+ℏω1​ ). It is shown that the mode with frequency ω3{{\omega }_{3}}ω3​ can localize in a squeezed state in the region of very strong coupling between the interacting modes, and the modes with frequencies ω1{{\omega }_{1}}ω1​ and ω2{{\omega }_{2}}ω2​ localize in states that are statistical mixtures of Fock states regardless of the value of the coupling coefficient
Abstract For the process of non-degenerate parametric decay in an optical cavity, where a photon with energy ћω3 decays into two photons with energies ћω2 and ћω1, where ћω3 = ћω2 + ћω1, the possibility of formation of entangled field states in terms of photon number variables between interacting modes of the optical system is investigated. Strong entangled states are shown between the modes with frequencies ω2 and ω1 in the case of strong coupling between interacting modes. In the case of weak and very strong coupling between these modes, relatively weak entangled states are formed. In the case of weak coupling, these modes are localized in entangled vacuum states with low values of quantum entropy. For modes with frequencies ω3 and ω1etangled states are formed only in the case of strong coupling.
For the process of non-degenerate parametric decay in an optical resonator, in which a quantum with energy ℏω3\hbar {{{{\omega }}}_{3}}ℏω3​ decays into two quanta with energies ℏω2\hbar {{{{\omega }}}_{2}}ℏω2​ and ℏω1\hbar {{{{\omega }}}_{1}}ℏω1​ , where ℏω3=ℏω2+ℏω1\hbar {{{{\omega }}}_{3}} = \hbar {{{{\omega }}}_{2}} + \hbar {{{{\omega }}}_{1}}ℏω3​=ℏω2​+ℏω1​ is the possibility of forming entangled states of the field by variables of the number of photons between interacting modes of the optical system is investigated. For this purpose, we investigated the dynamics of normalized correlation functions of fluctuations of the number of photons of interacting modes using a Monte–Carlo method. It is shown that in the stationary interaction limit, the value of these correlation functions strongly depends on the value of the coupling coefficient between the modes. It is shown that strong entanglement of the state of three interacting field modes is obtained in the case of weak coupling between the modes. In the case of extreme coupling between them, the value of the correlation function of fluctuations of the number of photons of the field modes tends to unity, because of which entanglement between the states of the modes disappears.
For the process of non-degenerate parametric decay in an optical cavity, where a photon with energy ћ3 decays into two photons with energies ћ2 and ћ1, where ћ3 = ћ2 + ћ1, the possibility of formation of entangled field states in terms of photon number variables between interacting modes of the optical system is investigated. It is shown that strongly entangled states are obtained between the modes with frequencies 2 and 1 in the case of strong coupling between interacting modes. In the case of weak and very strong coupling between these modes, relatively weak entangled states are formed. In the case of weak coupling, these modes are localized in entangled vacuum states with low values of quantum entropy. For modes with frequencies 3 and 1 entangled states are formed only in the case of strong coupling.
For the process of nondegenerate parametric decay in an optical cavity in which a photon with energy ℏω3\hbar {{{{\omega }}}_{3}}ℏω3​ decays into two photons with energies ℏω2\hbar {{{{\omega }}}_{2}}ℏω2​ and ℏω1\hbar {{{{\omega }}}_{1}}ℏω1​ , where ℏω3=ℏω2+ℏω1,\hbar {{{{\omega }}}_{3}} = \hbar {{{{\omega }}}_{2}} + \hbar {{{{\omega }}}_{1}},ℏω3​=ℏω2​+ℏω1​, the possibility of forming entangled field states of the interacting modes of the optical system based on the variable of quadrature amplitudes is studied. It is shown that strongly entangled states are obtained between the modes with the frequencies ω2{{{{\omega }}}_{2}}ω2​ and ω1{{{{\omega }}}_{1}}ω1​ in the case of a very strong coupling of the interacting modes. Weakly entangled states are formed in the case of strong coupling between modes, and no entanglement of these modes occurs in the case of weak coupling. In the optical system, the entangled states by the variable of quadrature amplitude do not form for the modes with frequencies ω3{{{{\omega }}}_{3}}ω3​ and ω1{{{{\omega }}}_{1}}ω1​ .
The quantum dynamics of the mean number of photons and quantum entropy of interacting modes, as well as the Wigner function of the stationary state of the fundamental mode and the third subharmonic mode has been investigated for the intracavity third-subharmonic generation. It is shown that the quantum dynamics of the system depends strongly on the nonlinear coupling coefficient between the modes. It is also demonstrated that, in the steady-state limit, depending on the intermodal coupling coefficient, the fundamental mode can be either in a pure coherent state, or in a squeezed state, or in a pure vacuum state. The third subharmonic mode in the subthreshold regime of generation of this mode is in the vacuum state. The Wigner function is squeezed over three sides of an equilateral triangle (squeezed vacuum). The quantum entropy of this state is nonzero. It is also shown that the third subharmonic mode, depending on the nonlinear coupling coefficient in the steady-state limit, can be localized in the three-component state with the same probability of detecting a field in each coherent component of the state and with the presence of quantummechanical interference between the state components. The mean number of photons in this state is smaller than unity. Depending on the nonlinear coupling coefficient, the third subharmonic mode can also be localized in the three-component state, which is a statistical mixture of three squeezed states.
A fully quantum treatment of the non-degenerate optical parametric oscillator in the extreme quantum limit of small photon numbers and very high nonlinear coupling strength is presented. When the nonlinear coupling constant becomes comparable with the cavity decay rate, the sharp threshold that is usually encountered disappears, a situation reminiscent of microlasers. Furthermore the output light exhibits unusual statistical properties such as strong super-Poissonian behaviour of the pump mode for (very) large coupling constants.