Shaping frequency-tunable single photons for quantum networking in waveguide QED

Abstract

The exchange of quantum information among nodes in a quantum network is one of the main challenges in modern technologies. Superconducting waveguide QED networks hold great potential for realizing distributed quantum computation, where distinct nodes communicate via itinerant single photons. Yet, different frequencies among the nodes restrict their applicability and limit scalability. Here we derive the controls required to shape single photons arbitrarily detuned with respect to their natural frequency, allowing thus for an on-demand and deterministic exchange of quantum information among frequency detuned nodes. We provide a theoretical framework, analyzing the properties of the controls for typical photon shapes, identifying operation regimes amenable for experimental realization. We then show how these controls enable frequency-selective quantum state transfer among non-resonant and distant nodes of a realistic network. In addition, we also provide a simple extension for remote entanglement generation between these nodes. The suitability and high-fidelity of these protocols is supported by numerical simulations, highlighting the novel networking possibilities unlocked when shaping frequency-tunable single photons.

Figures (7)

• Figure 1: Schematic representation of the frequency detuned single-photon emission $\gamma(t)$ via the control pulse $\tilde{g}(t)$ between states $| a \rangle$ and $| b \rangle$, where the latter decays at rate $\kappa$ into a dark state $| c \rangle$ emitting a photon $\gamma(t)$. By means of a time-dependent control of the phase and amplitude of $\tilde{g}(t)$, it is possible to emit a single-photon with temporal shape $\gamma(t)$, arbitrarily detuned with respect to its natural frequency. See main text for details.
• Figure 2: Required controls $g(t)$ to generate a detuned photon $\gamma(t)=\sqrt{\kappa/(4\eta)}{\rm sech}(\kappa t/(2\eta))$ (see Eqs. \ref{['eq:gt']}-\ref{['eq:varphiteta1']}). Panels (a) and (b) show the divergent behavior of the control when $\eta=1$, in amplitude and phase, respectively, for $\delta=-\kappa$ (solid red) and $\delta=-2\kappa$ (dashed blue). Note the logarithmic scale in panel (b) to better visualize the exponential scaling, that occurs for $\kappa t\gtrsim 0$. Panels (c) and (d) show again the control, amplitude and phase, respectively, for $\eta=3/2$ (solid red) and $\eta=2$ (dashed blue) for a fixed detuning $\delta=-2\kappa$. The horizontal (gray) lines in (c) correspond to the maximum value for both cases, $|g(t,\delta,\eta)|\approx |\delta|/\sqrt{\eta-1}$. See main text for further details.
• Figure 3: (a) Photon-emission infidelity $1-F_e(\tau)$ for a finite emission time $\tau$ as a function of the rescaled time $\kappa \tau/\eta$. Solid (black) line corresponds to the theoretical expression $1-F_e(\tau)=1-\tanh^2(\kappa \tau/(4\eta))$, while points show the numerically-computed infidelity for different photon detuning $\delta$ and reduced bandwidth $\eta$. Panel (b) shows the impact on the photon-emission infidelity as a function of the maximum allowed driving amplitude $g_m$ rescaled by $\delta$, for a fixed time $\kappa \tau=40$, and $\eta=1.25$ (orange), $\eta=2$ (blue) and $\eta=3$ (green). Solid (dashed) lines correspond to $\delta=2\kappa$ ($\delta=4\kappa$). (c) Spectra $|G(\omega,\delta,\eta)|^2$ of the controls for $\delta=4\kappa$, rescaled to their maximum, for three cases $\eta=3/2$ (orange circles), $2$ (blue diamonds) and $3$ (green triangles). Note the low-frequency components for $\eta=2$, and central frequencies located at $\omega=\delta(\eta-2)/(\eta-1)$. (d) Colormap of the difference between filtered with $\omega_{co}=10\kappa$ and ideal controls quantified by $\log_{10}S$ (cf. Eq. \ref{['eq:S']}) as a function of $\delta$ and $\eta$. The colormap encodes the scale of $\log_{10}S$, i.e. $S$ ranging from $10^{-5}$ (blue) to $1$ (red).
• Figure 4: (a) Schematic representation of the waveguide QED network, involving two nodes, A and B, and the quantum link of length $L$ connecting them. Node A involves a qubit $q_0$ and a resonator $c_0$ that is coupled to the quantum link. Node B consists of two qubits, $q_1$ and $q_2$, each interacting to its resonator, $c_1$ and $c_2$, respectively, that mediates the interaction with the quantum link. Each of the qubits is subject to a time-dependent coupling $g_i(t)$ with its resonator, allowing for an on-demand emission and absorption of single photons. (b) Frequencies $\omega_i$ for each of the three qubits and resonators in the network, such that $\omega_{1,2}=\omega_0+\delta_{1,2}$, being $\delta_{1}$ ($\delta_2$) the detuning of $q_1,c_1$ ($q_2,c_2$) with respect to $q_0$ and $c_0$. The arrows indicate the emission process of a detuned single photon from $q_0$ to match the frequency of either $q_1$ (blue) or $q_2$ (red) by means of a suitable control $g(t,\delta_{1,2},\eta)$. See main text for further details.
• Figure 5: (a) Emission ($g_0(t)$) and absorption ($g_{1,2}(t)$) controls for $\delta_{1}=-\delta_{2}=\kappa$ to produce a photon from $q_0$ shifted by $\delta_1$, i.e. targeting $q_1$. Panel (b) shows the dynamics of the qubit populations, $|q_i(t)|^2$ for a quantum state transfer protocol corresponding to (a). Vertical gray dashed lines indicate the emission and absorption times, at $\kappa t=0$ and $\kappa t\approx 30$, delayed by the propagation time of the photon through the waveguide $t_p$. Panel (c) shows the quantum state transfer infidelity $1-F_{i,qst}$ as a function of the frequency detuning between $q_1$ and $q_2$ for $q_0\to q_1$ (blue circles) and $q_0\to q_2$ (red squares), symmetrically shifted from $\omega_0$, i.e. $\Delta=\delta_1-\delta_2=2\delta$, including photon loss and $T_1$ noise. Horizontal dashed line corresponds to $p_{loss}=6.9\cdot 10^{-3}$. Panel (d) shows $F_{1,qst}$ for a fixed detuning of the qubit $q_1$ of $\delta_1=2.5\kappa$ as a function of the emitted detuned photon $\delta_c$, i.e. $g_0(t)=g(t,\delta_c,\eta)$, both for the simulated results (blue points) and theoretical expression $O(\delta_c-\delta_1)$ in Eq. \ref{['eq:O']} (orange dashed). For $\delta_c=\delta_1$, the transfer is nearly perfect (corresponding to $\Delta=5\kappa$ in panel (c)), while it quickly drops to zero otherwise. The parameters in all panels are $\kappa=2\pi\times 30$ MHz, $\eta=2$ and $L=30$ m with $\omega_0=2\pi\times 8.5$ GHz.