# Low-Noise Frequency Translation of Single Photons via Four Wave Mixing Bragg Scattering

Abstract

We present a single-photon frequency translation setup based on Four Wave Mixing Bragg-Scattering in fiber, able to achieve simultaneously close to unitary conversion while maintaining very low-noise.

# Introduction

The creation of a scalable quantum-enabled machine, e.g. a quantum computer or a quantum communication system, requires the development of many components, such as sources, memories, processing elements, measurement devices and a network able to inter-connect the different resources (Shapiro 2002, Kimble 2008). Optical connections are indeed good candidates for the transmission of quantum information, in the form of flying qubits, as single photons or other nonclassical states, that can travel undisturbed over long distances. The major obstacle to such scheme is that often different elements work at different energies, and it may be difficult to link together at a fixed optical frequency.

Frequency translation (Kumar 1990) provides a mean to shift and in general modify the carrier frequency of an optical quantum state while preserving the other quantum features (such as coherence (Tanzilli 2005) and entanglement (Ramelow 2012)), thus linking the different resources, e.g. IR flying qubits with efficient silicon detectors (Albota 2004, Langrock 2005, Vandevender 2004, Ates 2012). More recently, frequency translation has emerged as a mean to directly manipulate the optical waveform in the full temporal and spectral space: using concepts borrowed from parametric time-lens, both temporal compression (Agha 2013) and magnification (Lavoie 2013) of quantum states can be implemented. In addition, using time-dependent conversion, it is possible to select and convert different temporal modes, realizing an optical pulse gate (Reddy 2014, Christensen 2015) that encode and decode high-dimensional qubits. While sum- and difference-generation in $$\chi^{(2)}$$ has been the base for many demonstrations, thanks to the high nonlinearity and ease of setup, optical frequency translation as been seen in other systems, including cross-phase modulation (Bradford 2012, Matsuda 2014), opto-mechanical hybrid systems (Hill 2012, Preble 2012), electro-optical modulation (Mérolla 1999), Alkali-vapor cells (Donvalkar 2014), microwave superconductor resonators (Zakka-Bajjani 2011), Diamond based atomic memories (Fisher 2016). Among parametric $$\chi^{(3)}$$ processes, Bragg Scattering is known to frequency translate quantum states without the addition of parametric noise (McKinstrie 2005). Follow up realizations include photonic crystal fiber (McKinstrie 2005, McGuinness 2010, Mejling 2012), highly nonlinear fiber (Gnauck 2006, Clark 2013, Krupa 2012) and SiN and S waveguides and resonators (Agha 2013, Li 2016, Bell 2016). This implementation supports translation across any span of energy, enabling conversion between frequencies in the same communication band and small tuning. In addition, $$\chi^{(3)}$$ implementations are compatible with integrated photonics.

Both $$\chi^{(2)}$$ and $$\chi^{(3)}$$ implementations have to grapple with technical noise, usually generated by the strong pump driving the nonlinear process. In the case of the other implementations conversion efficiency close to 100% with limited residual noise has been obtained, for Bragg-Scattering, conversion at unitary efficiency was only obtained at the expenses of a signal polluted by large Raman noise (Clark 2013). In this work, thanks to an optimal choice of medium and operating wavelength, we show Four Wave Mixing Bragg Scattering (FWM-BS) setup that performs frequency translation at very low noise regime, with almost unity efficiency on weak coherent states and single photons alike.

\label{fig:cartoon} a) Energy levels picture of Four-Wave Mixing Bragg Scattering: two strong pump with angular frequency difference $$\Delta \Omega$$ drive the conversion between a signal and an idler field with the same frequency difference. b) Interacting fields in respect of the dispersion (green line) of the nonlinear medium: in the first approximation, fields are chosen to be symmetrical in respect of the zero-dispersion wavelength (ZDW).

# Four-wave mixing Bragg Scattering

FWM-BS interactions are driven by two strong pump fields, whose angular frequency separation $$\Delta\omega$$ defines the amount of frequency translation between signal and idler (see fig. \ref{fig:cartoon}). The efficiency of the translation process is $$\eta=r^2/k_{bs}^2 \sin^2(k_{bs} L)$$, where $$r = 2\gamma P$$ is the coupling strength term, $$k_{bs} = \sqrt{\kappa^2+r^2}$$ the scattering wavevector, and $$\gamma$$, $$L$$ and $$P$$ respectively the medium nonlinearity, the interaction length and the pump power. $$\kappa = \beta(\omega_s) - \beta(\omega_i) - \beta(\omega_1) + \beta(\omega_2)$$ is the phasematching term: it is convenient to expand the wavevector $$\beta(\omega)$$ in respect of the zero dispersion frequency $$\omega_{ZDW}$$ of the nonlinear medium (i.e. $$\beta^{(2)}(\omega_{ZDW}= 0$$) so that $$\beta(\omega) = \beta^{(3)}/6 (\omega-\omega_{ZDW})^3 + \beta^{(4)}/24 \cdot (\omega- \omega_{ZDW})^4+...$$, and to introduce the average pump frequency $$\Delta \Omega = (\omega_1 + \omega_2) /2 - \omega_{ZDW}$$ and the signal offset $$\epsilon$$ so that the frequency of signal and idler can be written as $$\omega_s = \epsilon -\Delta\Omega + \Delta\omega/2$$ and $$\omega_s = \epsilon -\Delta\Omega - \Delta\omega/2$$.

The phasematching now reads \begin{aligned} \kappa &= \frac{\beta^{(3)}}{6} \left[ {\left(\epsilon -\frac{\Delta\omega}{2} -\Delta\Omega\right)}^3 - {\left(\epsilon +\frac{\Delta\omega}{2} -\Delta\Omega\right)}^3 + {\left( \frac{\Delta\omega}{2} +\Delta\Omega\right)}^3 - {\left(-\frac{\Delta\omega}{2} +\Delta\Omega\right)}^3 \right] + \\ &+ \frac{\beta^{(4)}}{24} \left[ {\left(\epsilon -\frac{\Delta\omega}{2} -\Delta\Omega\right)}^4 - {\left(\epsilon +\frac{\Delta\omega}{2} -\Delta\Omega\right)}^4 + {\left( \frac{\Delta\omega}{2} +\Delta\Omega\right)}^4 - {\left(-\frac{\Delta\omega}{2} +\Delta\Omega\right)}^4 \right] \end{aligned} and simplifies to $\label{eq:ph} \kappa = \frac{\beta^{(3)}}{6} \left[ 3 \Delta\omega \epsilon (\epsilon-2\Delta\Omega) \right] + \frac{\beta^{(4)}}{24} \Delta\omega \left[ 2\Delta\Omega (\Delta\omega-4\Delta\Omega^2+3\epsilon(\Delta\Omega-\epsilon)) - \epsilon (\Delta\omega^2-4\epsilon^2) \right]$

In the straightforward case of $$\epsilon = 0$$, i.e., the fields two and two are symmetrical in respect of $$\omega_{ZDW}$$ (as in figure \ref{fig:cartoon}b ), the $$\beta^{(3)}$$ term (as well as all the other odd terms of $$\kappa$$) cancels out, leaving only the contribution from higher order dispersion. In the approximation $$\Delta\Omega \gg \Delta\omega \gg \epsilon$$ we obtain a simpler expression for the process momentum conservation $\kappa(\epsilon) L \simeq \beta^{(3)} L \Delta\omega \Delta\Omega \epsilon + \beta^{(4)}L /24 (8\Delta\omega\Delta\Omega^3) = (\epsilon + \Delta\Omega)/\delta\omega_{bs}$ in which we can identify the process acceptance-bandwidth $$\delta\omega_{bs} = (\beta^{(3)} L \Delta\Omega\Delta\omega)^{-1}$$, and the frequency separation from symmetric point $$\delta\epsilon = \frac{\beta^{(4)}}{3 \beta^{(3)}} \Delta\Omega^2$$ due to higher-order dispersion (Provo 2010). One prominent feature of FWM-BS, already noticed in (Inoue 1994, Marhic 1996) is highlighted by equation \ref{eq:ph}, that is translation for any given pair of signal and idler frequency can be exactly phasematched by choosing the appropriate pumps: this gives the flexibility of tuning the parameters of the interaction without modifying the nonlinear medium dispersion.