Weak-Field Expansion: A Time-Closed Solution of Quantum Three-Wave Mixing

Abstract

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 fidelity of quantum state-transfer from one optical mode to another using sum/difference frequency generation, due to the quantum properties of the strong driving field.

Figures (4)

• Figure 1: Second order correction to the pump field at the output of an SU(1,1) interferometer: (a) SU(1,1) concept with marking of the pump field at the input and output. (b) The number of photons depleted/added to the pump $\Delta N_p\!=\!\bigl\langle\hat{a}_{p2}^{\dagger}\hat{a}_{p2}\bigr\rangle\!-\!|\alpha_{p}|^2$ due to PDC/SFG; and (c) The phase of the output pump field, as a function of the parametric gain $g$ (radius) and of the total phase of the input fields $\Delta\phi\! =\! \phi_s\!+\!\phi_i\!-\!\phi_p$ (angle), plotted in polar coordinates; (d) and (e) are section plots along the $\pi$-$0$ phase-line (real axis) of (a), and along the $3\pi/2$-$\pi/2$ phase line (imaginary axis) of (b), respectively (negative $g$ on the $x$-axis indicate a $\pi$ phase). We assume an ideal SU(1,1) interferometer (no internal loss) with equal parametric gain $g$ in the two nonlinear media.
• Figure 2: The non-classical nature of SFG with squeezed input: We compare the efficiency of stimulated SFG $\Delta N_p/N_{in}$ (the number of SFG photons that are added to the pump relative to the number of input signal and idler photons) as the function of parametric gain $g$, under two input conditions: (1) ideally squeezed input (blue), as generated by SPDC with the same gain, i.e. SU(1,1) interference at $\pi$ phase; (2) equal unentangled signal and idler (red), upper limit for the conversion efficiency with coherent quantum input, or the classical prediction of stimulated SFG. Evidently, the SFG with an entangled input remains ideal for any gain$g$, while for a coherent unentangled input the SFG efficiency vanishes for low $g$, as classically expected.
• Figure 3: Quanrtm state-transfer: third order corrections (a) The optimal imaginary gain of classical state transfer for varying input intensities (the $x$-axis is $m\!=\!{\bigl\langle\hat{N}_{p,s}\bigr\rangle}/{{|\alpha_i|}^2}$ — the photon-number ratio between the input state and the strong idler). (b) shows the corresponding quantum correction to the average number of transferred photons $\Delta N\!=\!N_{p,s}(t)\!-\!N_{s,p}(0)$, and (c) shows the correction to the variance of the photon-number $\Delta\operatorname{Var}{\hat{N}_{s,p}}\!=\!\operatorname{Var}\hat{N}_{s,p}(t)\!-\!\operatorname{Var}\hat{N}_{p,s}(0)$, indicating that the variance of the output depends not only on the $m$ ratio, but also on the absolute intensity of the input state. The blue (red) lines indicate signal-to-pump (pump-to-signal) transfer. The calculations here assumed an input of a weak coherent state.
• Figure 4: Quantum state transfer: The quantum correction to the variance of the transferred photon-number in the target state as a function of the input intensity under fixed driving idler — ${|\alpha_i|}^2={10}^4$ (solid lines) and ${|\alpha_i|}^2={10}^5$ (dashed lines). The input state of the pump/signal source is assumed coherent. The two plots for the two cases under ${|\alpha_i|}^2={10}^5$ almost overlap.