Scattering of two photons on a quantum emitter in a one-dimensional waveguide: exact dynamics and induced correlations

Abstract

We develop a wavefunction approach to describe the scattering of two photons on a quantum emitter embedded in a one-dimensional waveguide. Our method allows us to calculate the exact dynamics of the complete system at all times, as well as the transmission properties of the emitter. We show that the nonlinearity of the emitter with respect to incoming photons depends strongly on the emitter excitation and the spectral shape of the incoming pulses, resulting in transmission of the photons which depends crucially on their separation and width. In addition, for counter-propagating pulses, we analyze the induced level of quantum correlations in the scattered state, and we show that the emitter behaves as a nonlinear beam-splitter when the spectral width of the photon pulses is similar to the emitter decay rate.

Limitations of two-level emitters as nonlinearities in two-photon controlled-phase gates

“…This is because it is for these widths that the nonlinearities are strongest and the required π phase shift can be generated, consistent with the results in Ref. [20]. 1 Furthermore, the optimal value of L in this nonlinear-scattering case is significantly lower than in the linear case.…”

“…(4) and (8), we see that a good gate performance requires |k| ˜ , which corresponds to pulses with a spectrum that is much narrower than the emitter linewidth. For spectrally broader photons for which ξ (k) extends beyond k ∼˜ , terms of higher order in k will have an influence and introduce chirping effects [19,20].…”

“…The idealized gate we consider uses the in-line nonlinearities of two two-level-emitters embedded in lossless waveguides. The fundamental operating principle of the gate relies on the saturability of a two-level emitter, which means that the phase imparted onto a photon or photons scattering on such an emitter depends on how many photons are present [19,20,37,38]. Although it has been shown that such a gate can never perform with perfect fidelity [38,39], the purpose of this work is to understand the limits and origins of its imperfections with a view towards improved future implementations.…”

Strong nonlinearity-induced correlations for counterpropagating photons scattering on a two-level emitter

“…A weaknonlinearity-induced four-wave mixing process is signified by the appearance of diagonal features, as one photon achieves a larger energy, while the energy of the other decreases. When the pulse and emitter linewidths are comparable (second row), the predicted strong directional correlation is induced [15], with the pulse profile being almost preserved. For spectrally broad pulses (third row), only the near-resonant part of the spectrum interacts with the emitter.…”

“…As such, identical input photons of a specific spectral lineshape are required. In the single-photon case, however, it is known that such finite-width photons experience changes in both their spectra and phase as a result of the scattering process [16,22], which is also known to be the case for two-photon scattering [15]. Thus, once a photon has passed through an optical gate, it is no longer identical to an input photon, which in turn may limit the effectiveness of subsequent gates.…”