Quantum State Transfer via Noisy Photonic and Phononic Waveguides

We describe a quantum state transfer protocol, where a quantum state of photons stored in a first cavity can be faithfully transferred to a second distant cavity via an infinite 1D waveguide, while being immune to arbitrary noise (e.g., thermal noise) injected into the waveguide. We extend the model and protocol to a cavity QED setup, where atomic ensembles, or single atoms representing quantum memory, are coupled to a cavity mode. We present a detailed study of sensitivity to imperfections, and apply a quantum error correction protocol to account for random losses (or additions) of photons in the waveguide. Our numerical analysis is enabled by matrix product state techniques to simulate the complete quantum circuit, which we generalize to include thermal input fields. Our discussion applies both to photonic and phononic quantum networks.

Figure 1

Quantum state transfer via a noisy waveguide. (a) QST where qubits are coupled directly with chiral coupling to a waveguide representing the quantum channel. (b) QST in a cavity QED setup, where atoms representing qubits are coupled to the waveguide with a cavity as mediator. (c) Fidelity $\mathcal{F}$ for QST of a qubit as a function of photon occupation $n_{\text{th}}$ representing a thermal noise injected into the waveguide for setups (a) and (b). For the protocol described in the text, setup (b) is robust to injected noise. (d) “Write” of a quantum state from cavity 1 to a temporal mode in the (noisy) waveguide, and “Read” back to cavity 2 as a linear multimode encoder and decoder with encoding [decoding] functions ${\kappa }_1(t)$ [${\kappa }_2(t)$] (see text).

Figure 2

Role of imperfections. (a) Effect on the fidelity of a finite transfer time $T=t_f-t_i$, and (b) of an imperfect timing of ${\kappa }_2(t)$. (c) Fidelity as a function of $\varphi$ for different $\beta$ factors and ${\kappa }_{\max }\tau \approx 0$ (see text). Solid lines: $n_{\text{th}}=0$. Dashed lines: $n_{\text{th}}=0.25$. The fidelity is maximal when $\varphi$ is a multiple of $\pi$. (d) By increasing ${\kappa }_{\max }\tau$ for $\varphi =0$ the fidelity decreases. (e) QEC with the setup subject to waveguide and cavity losses. (f) QEC with the setup coupled to a reservoir with photon occupation $n_{\text{th}}^{\prime }=1$. Black: no error correction. Red: correction against single photon losses. Blue: correction against single photon losses or additions. Solid lines: $n_{\text{th}}=0$, ${\kappa }^{\prime }=0$. Full circles: $n_{\text{th}}=0.5$, ${\kappa }^{\prime }=0$. Empty circles: $n_{\text{th}}=0$, ${\kappa }_f=0$.

Figure 3

QST in noncascaded systems. (a) QST in a closed system with two cavities coupled to a finite waveguide. (b) Fidelity as a function of the cavity nonlinearities $\chi$ and for different initial occupation of the waveguide. The fidelity approaches unity in the linear limit $\chi \to 0$.