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arXiv:2201.01372 · Physical Review A

Pump depletion in optical parametric amplification

Wanli Xing, Timothy C. Ralph

Published Feb 14, 2023• 12 citations
arXivHTMLPDF

Abstract

We derive analytic solutions for Heisenberg evolution under the trilinear parametric Hamiltonian which are correct to the second order in the interaction strength but are valid for all pump amplitudes. The solutions allow pump depletion effects to be incorporated in the description of optical parametric amplification in experimentally relevant scenarios and the resulting phenomena to be rigorously described.

DOI
10.1103/physreva.107.023712

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We derive analytic solutions for Heisenberg evolution under the trilinear parametric Hamiltonian which are correct to the second order in the interaction strength but are valid for all pump amplitudes. The solutions allow pump depletion effects to be incorporated in the description of optical parametric amplification in experimentally relevant scenarios and the resulting phenomena to be rigorously described.
We derive analytic expressions for observable dynamics under the fully quantized trilinear Hamiltonian, governing quantum three-wave mixing and parametric amplification. Starting from a coherent pump and vacuum signal and idler, the calculation reorganizes the Heisenberg expansion into resummed depletion and return profiles, yielding directly evaluable expressions for all 1st-, 2nd- and 3rd-order correlations. The construction uses no parametric approximation or fitted parameters. Representative numerical benchmarks show errors of order 10−1210^{-12}10−12 throughout the entire pump depletion, and below 10−310^{-3}10−3 across dozens of depletion-and-back-conversion cycles, up to the first total collapse of the oscillation. The first-depletion treatment additionally provides 4th-order correlations, accommodates bright squeezed-pump inputs and reveals phenomena important to future optical quantum technologies. The framework can be extended to higher moments.
A quantum-statistical treatment of degenerate (single-mode) parametric amplification is given in which the pump is allowed to have time-dependent amplitude and phase. This relaxes earlier restrictions of a constant pump in exact resonance with the signal. Solutions are obtained for the case where the ratio of the time derivative of the pump phase to the coupling strength between pump and signal is constant. This includes the special case of simple offresonance, where the pump has a constant frequency differing from twice the signal frequency by a given amount (and has constant amplitude). The Wigner distribution for the signal mode is obtained (the PPP distribution does not in general exist as a well-defined function beyond a finite time interval for this process) and is used to determine the time development of the expectation values of various field operators of interest, such as field strengths and fluctuations. A critical point exists in the solution for the photon annihilation operator such that the behavior of the solution and the resulting quantities which it determines is divided into an exponential region and an oscillatory region. The critical point defines, from the expectation value for the photon number, the condition for amplification. For the case of simple off-resonance, this gives the frequency range of amplification. A calculation without using the rotating-wave approximation is given in the Appendix which shows that, to first order, the same range is obtained as when the approximation is used. The effect of damping is given, showing, e.g., how it slows the growth rate of the signal (in the region of amplification) and how it narrows the frequency range of amplification. The effect of a time-dependent pump (and damping) on the breakdown of the PPP distribution is also described.
Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.
We present a systematic derivation of the Heisenberg evolution of a trilinear bosonic Hamiltonian system in presence of a strong drive beyond the standard approximation of a classical, undepleted driving field. We employ a perturbative expansion of the Hamiltonian propagator in orders of the input field amplitudes, as opposed to the standard Baker-Campbell-Hausdorff (BCH) expansion of the propagator in orders of time. Our method automatically provides time-closed expressions; and converges considerably faster than BCH, especially in the regime of high parametric gain because the small parameter it uses is natural to the problem. We obtain the well-known quantum solution for optical parametric amplification of down-conversion simply as the first order of the expansion, and present the rigorous procedure to derive higher order corrections one by one. To demonstrate the utility of higher corrections, we discuss the 2nd order correction to the pump field as an ideal detector of time-energy entanglement in parametric down-conversion. We also use the 3rd order correction to calculate the limits on the achievable squeezing with parametric amplification and on the fidelity of quantum state-transfer from one optical mode to another with sum/difference frequency generation, due to the quantum properties of the strong driving field.
An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).