Passive, deterministic photonic conditional-phase gate via two-level systems

Abstract

Photons are one of the best carriers of quantum information, as they interact only weakly with the environment, and photon processes are fast and efficient. Despite these advantages, two main issues have prevented alloptical quantum computing from being realized as effectively as computing with ions, trapped atoms, or superconducting qubits: true on-demand single photon sources are difficult to create and photon-photon interactions are challenging to engineer. In this paper we propose a realistic means of circumventing this second challenge and demonstrate how two level emitters (TLEs) coupled to a one-dimensional waveguide can, in principle, implement a passive, near-deterministic conditional phase (CPHASE) gate between two photons. There have been many proposals to overcome this second challenge. Many schemes of all-optical QIP (such as [1]) only work probabilistically, requiring a substantial overhead. Other proposals require active elements or control of an optical medium [2–4] but this typically makes them difficult to scale up to large computations. Schemes based on temporary storage of one of the photons are either complicated by the need for several steps [5], or require quantum memories for intermediate storage [6]. Ideally, what one would want is to simply send two photons flying through a nonlinear medium and have them exit with a useful, conditional phase, as originally proposed in [7]. However, existing, conventional media do not exhibit sufficiently strong optical nonlinearities, and may be too “noisy” for quantum information processing applications [8], while spectral entanglement has also been identified as a potential issue in these schemes [9]. A recent important step in this direction was provided by Brod and Combes in [10, 11], where they showed that an array of two level systems can, in principle, allow two photons to conditionally interact without becoming spectrally entangled. Central to their scheme is the idea that the photons must propagate in opposite directions through the array, requiring either a perfect chiral coupling between the emitters and the waveguide, or the use of optical circulators at every step. Additionally, their proposal required some way to have the TLEs interact while keeping the photons physically separate. Here we show that an array of two-level atoms coupled to an ordinary (non-chiral) waveguide can, in principle, perform the CPHASE operation between two counterpropagating, single-photon wavepackets. Our scheme relies on the existence of transmission resonances in the interaction of a single photon with a pair of TLEs, which we pointed out in [12] and which have been interpreted as Fano resonances by other authors ([13]; see also [14– 16]). We showed in [12] that, under the right conditions, these transmission windows persist in the nonlinear regime where two counterpropagating photons interact with the pair of TLEs, so the photons are transmitted with near-unit probability, but they pick up a nontrivial phase as they do so. Ideally, a conditional phase gate would impart a phase φ1 to the states |0〉|1〉 and |1〉|0〉 (where 1 and 0 refer to the presence or absence of a photon) and a phase φ2 to the state |1〉|1〉. As long as Φ = φ2 − 2φ1 6= 0, this gate, together with single photon gates, would enable universal quantum computation. In practice, we desire Φ to be as large as possible while maintaining a high fidelity, as discussed below. In our scheme, as we show here, Φ = π is possible for an array of pairwise interacting atoms, and Φ = π/2 for a system without interactions. Regarding the issue of fidelity, we note that (in the same way as [10]), our gate will distort the spectrum of a single photon. As pointed out by Brod and Combes, this can be overcome by ensuring that at every step in the computation all photons are distorted in the same way. We then define the fidelity of the two-photon operation relative to the product state of two independent singlephoton transmissions through the array of N pairs of scatterers. Formally, we write √ Fe = 〈Target|ψScattered〉 (1)