Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

Abstract

While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the effect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated optical fields has never been questioned before. Here, we provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-$\Lambda$ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.

Correction of v1

References in corpus (8)

• Narrow-Band Biphoton Generation near Atomic Resonance
• Anti-Parity-Time Symmetric Optical Four-Wave Mixing in Cold Atoms
• Quantum squeezing and sensing with pseudo anti-parity-time symmetry
• Narrowband biphoton generation in the group delay regime
• Four-wave mixing in a non-degenerate four-level diamond configuration in the hyperfine Paschen-Back regime
• Quantum-State Purity of Heralded Single Photons Produced from Frequency-Anti-Correlated Biphotons
• Nonlinear power dependence of the spectral properties of an optical parametric oscillator below threshold in the quantum regime
• Quantum correlations from dynamically modulated optical nonlinear interactions

Cited by in corpus (1)

• Theory of Symmetry-Protected Two-Photon Coherence