Entanglement of photonic modes from a continuously driven two-level system

Abstract

The ability to generate entangled states of light is a key primitive for quantum communication and distributed quantum computation. Continuously driven sources, including those based on spontaneous parametric downconversion, are usually probabilistic, whereas deterministic sources require accurate timing of the control fields. Here, we experimentally generate entangled photonic modes by continuously exciting a quantum emitter - a superconducting qubit - with a coherent drive, taking advantage of mode matching in the time and frequency domain. Using joint quantum state tomography and logarithmic negativity, we show that entanglement is generated between modes extracted from the two sidebands of the resonance fluorescence spectrum. Because the entangled photonic modes are perfectly orthogonal, they can be transferred into distinct quantum memories. Our approach can be utilized to distribute entanglement at a high rate in various physical platforms, with applications in waveguide quantum electrodynamics, distributed quantum computing, and quantum networks.

Keywords: Quantum information; Qubits.

Fig. 1. Experimental implementation for entangled-photon generation based on a superconducting circuit.

a False-color optical micrograph of the device. A transmon qubit (orange) is capacitively coupled to a waveguide (red). The coupling element visible to the left of the transmon is not used in this work. b Schematic representation of the device and the measurement setup. ${\widehat{a}}_{\mathrm{in}}$ and ${\widehat{a}}_{\mathrm{out}}$ represent the input and output modes of the emitter’s field, respectively. c Temporal mode matching. The emitter is continuously driven at its frequency ω_ge (orange), and its emitted radiation is recorded as a time trace (red). The two insets represent the temporal filters f₁(t) (blue) and f₂(t) (green) applied on the time trace to match the two propagating modes.

Fig. 2. Second-order moment ⟨a^1†a^1⟩ of a single propagating mode under varying drive conditions.

a $⟨{\widehat{a}}_1^{†}{\widehat{a}}_1⟩$ of the output mode as a function of Rabi frequency $\mathrm{\Omega }$, with the template-matching duration fixed at $T=100\text{ns}$. b $⟨{\widehat{a}}_1^{†}{\widehat{a}}_1⟩$ as a function of template-matching duration $T$, with $\mathrm{\Omega }$ fixed at $4.04\mathrm{\Gamma }$. In both panels, each curve is vertically offset by 0.6 for clarity. The filled circles are the measured data, while the black solid lines are the simulation results. The red vertical lines mark the positions of $\pm \mathrm{\Omega }$, which match the location of the side peaks of the corresponding curves.

Fig. 3. Moments and correlations of temporally matched modes.

a–h Moments up to second order for the temporal modes defined in Eq. (3) as a function of frequency detunings Δ_k = ω_k − ω_ge, where k = 1, 2. The upper row shows the data measured from the experiments. The lower row shows the 2D maps of the moments calculated from the simulation. The 2D map of the second-order moments $⟨{\widehat{a}}_1^{†}{\widehat{a}}_1⟩$ and $⟨{\widehat{a}}_2^{†}{\widehat{a}}_2⟩$ are normalized to have the same maximum value as the simulation, and the other moments are scaled accordingly. i, j The real and imaginary parts of moments up to fourth order at the frequencies (Δ₁, Δ₂) = (Δ₋, Δ₊). The blue bar is the measured data, while the black wireframe is the simulation. The error bar for each measured moment is obtained by splitting the data into 20 segments, calculating the moments for each segment, and obtaining the standard deviation over them.

Fig. 4. Entangled temporal modes.

Density matrix from the joint quantum state tomography and the obtained logarithmic negativity. a Comparing the real part of the reconstructed and simulated density matrices in Fock space up to N = 5, while only the selected lower-order components satisfying N≤3 are shown. The colored bars show the measurement result, and the black wireframes are the simulated prediction. The imaginary components of both the simulated and reconstructed density matrices, which are not shown, are less than 0.015 across all elements of the matrices. b The logarithmic negativity $E_{\mathcal{N}}$ reconstructed from the 27 measured moments over the frequency ranges Δ₁ ∈ [−40, −10] MHz and Δ₂ ∈ [10, 40] MHz. There are seven not-a-number points, shown in white, due to the optimizer failing to meet the standard deviation constraints of the moments at these frequency points during reconstruction. c The numerically simulated logarithmic negativity $E_{\mathcal{N}}$ over the same frequency range. d $E_{\mathcal{N}}$ reconstructed from the 27 simulated moments over the same frequency ranges. This distribution matches (c) if reconstructed using all combinations of simulated moments up to N = 5 (see “Methods”, “Joint Quantum state tomography”).

Fig. 5. State overlap of the joint quantum state tomography.

a State overlap between reconstructed density matrices (using 27 measured moments) and simulated density matrices, over the frequency ranges Δ₁ ∈ [−40, −10] MHz and Δ₂ ∈ [10, 40] MHz, utilizing least-squares optimization. b State overlap over the same frequency ranges, utilizing compressed-sensing optimization. Note the white points indicating not-a-number values due to optimization failures during reconstruction [see explanation under Fig. 4(b)].

Fig. 6. Purity of density matrices.

a Purity obtained from the reconstructed density matrices using 27 measured moments, utilizing least-squares optimization, over the frequency ranges Δ₁ ∈ [−40, −10] MHz and Δ₂ ∈ [10, 40] MHz. b Purity obtained from the same moments as in (a), but with compressed-sensing optimization. Note the white points indicating not-a-number values due to optimization failures during reconstruction [see explanation under Fig. 4(b)]. c Purity obtained from numerical simulations.

Fig. 7. The logarithmic negativity EN.

a $E_{\mathcal{N}}$ reconstructed using all 325 simulated moments for a Fock space cutoff at N = 5, utilizing least-squares optimization, over the frequency ranges Δ₁ ∈ [−40, −10] MHz and Δ₂ ∈ [10, 40] MHz. b $E_{\mathcal{N}}$ reconstructed using the same simulated moments as in (a), but with compressed-sensing optimization, over the same frequency range. c $E_{\mathcal{N}}$ obtained from numerical simulation, over the same frequency range.