Photon-photon interactions in dynamically coupled cavities

Abstract

We study theoretically the interaction between two photons in a nonlinear cavity. The photons are loaded into the cavity via a method we propose here, in which the input/output coupling of the cavity is effectively controlled via a tunable coupling to a second cavity mode that is itself strongly output-coupled. Incoming photon wave packets can be loaded into the cavity with high fidelity when the timescale of the control is smaller than the duration of the wave packets. Dynamically coupled cavities can be used to avoid limitations in the photon-photon interaction time set by the delay-bandwidth product of passive cavities. Additionally, they enable the elimination of wave packet distortions caused by dispersive cavity transmission and reflection. We consider three kinds of nonlinearities, those arising from χ (2) and χ (3) materials and that due to an interaction with a two-level emitter. To analyze the input and output of few-photon wave packets we use a Schrödingerpicture formalism in which travelling-wave fields are discretized into infinitesimal time-bins. We suggest that dynamically coupled cavities provide a very useful tool for improving the performance of quantum devices relying on cavity-enhanced light-matter interactions such as single-photon sources and atom-like quantum memories with photon interfaces. As an example, we present simulation results showing that high fidelity two-qubit entangling gates may be constructed using any of the considered nonlinear interactions.

Controlled-Phase Gate Using Dynamically Coupled Cavities and Optical Nonlinearities

“…The output is therefore in a mixed state but we only calculate the dynamics of the zero-loss subspace where the output states above are not normalized and the fidelity in Eq. (1) is a lower bound [12].…”

“…We refer to Ref. [12] for detailed derivations of all the equations of motion and input-output relations used here.…”

“…In Ref. [12] we derived solutions for Λ (k) n that enable complete absorption of a single photon with an arbitrary wave packet or emission of an arbitrary output, where k refers to a χ (k) material. The solutions differ due to crossphase modulation imparted on cavity modes a and b by the control fields only in χ (3) materials.…”

“…3 also plots the error in the conditional one-photon state fidelity defined by F 1 ≡ F 1 / 1 |1 . F 1 may be understood as the probability of the input and output states being identical given there was no loss because it corresponds to the fidelity calculated using the re-normalized state |1 / 1 |1 [12]. The ideal scenario for lossy cavities is that the output wave packet is a scaled version of the input, ξ out (t) = √ ηξ in (t − T ).…”

“…For a given loss rate, γ L , there is a corresponding value of η from which Λ (k) n is calculated to achieve ξ out (t) ≈ √ ηξ in (t−T ), see Ref. [12] for details. Since the conditional fidelity by definition is independent of the scaling factor η, we expect it to be negligibly dependent on loss so that F 1 ≈ F 1 (γ L = 0), which is confirmed in Fig.…”