Correlations and Entanglement of Microwave Photons Emitted in a Cascade Decay

We use a three-level artificial atom in the ladder configuration as a source of correlated, single microwave photons of different frequency. The artificial atom, a transmon-type superconducting circuit, is driven at the two-photon transition between ground and second-excited state, and embedded into an on-chip switch that selectively routes different-frequency photons into different spatial modes. Under continuous driving, we measure power cross-correlations between the two modes and observe a crossover between strong antibunching and superbunching, typical of cascade decay, and an oscillatory pattern as the drive strength becomes comparable to the radiative decay rate. By preparing the source in a superposition state using an excitation pulse, we achieve deterministic generation of entangled photon pairs, as demonstrated by nonvanishing phase correlations and more generally by joint quantum state tomography of the two itinerant photonic modes.

Figure 1

Photon source and microwave switch. (a) Simplified scheme of the measurement setup. The photon source (PS) is coherently driven from input port 1. Input port 4 is used to characterize the scattering properties of the switch. The output modes $\widehat{e}$ and $\widehat{f}$ are amplified by Josephson parametric dimers (JPDs). The two quadratures of the amplified signals are measured by double heterodyne detection and recorded by an analog-to-digital converter (ADC). (b) False-color micrograph of the sample, illustrating the PS (blue) and the key components of the switch, two $\pi /2$ hybrid couplers (red) and two tunable resonators (green). (c) Closeup micrograph of the PS, a transmon asymmetrically coupled to input and output lines, and level scheme indicating the relevant transitions. The transmon is driven at the two-photon transition ${\omega }_{gf}/2$ with rate ${\mathrm{\Omega }}_{\mathrm{gf}}$ and decays by emitting photons at frequencies ${\omega }_{ef}$ and ${\omega }_{ge}$ and rates ${\mathrm{\Gamma }}_{ef}$ and ${\mathrm{\Gamma }}_{ge}$, respectively. (d) Characterization of the switch via port 4: transmittance to ports 3 ($|S_{34}{|}^2$) and 2 ($|S_{24}{|}^2$) vs frequency $\nu$. The transition frequencies of the transmon, ${\nu }_{ge}$ and ${\nu }_{ef}$, are indicated by dashed lines.

Figure 2

Inelastic scattering under continuous excitation. (a) Power spectral density (PSD) of the radiation emitted by the source, driven at ${\omega }_{\mathrm{gf}}/2$ and low power: theory (solid line) and combined measured traces (colored dots) from the two modes. (b) Measured power spectral densities in modes $\widehat{f}$ (left) and $\widehat{e}$ (right) for varying input powers, $P_{\mathrm{in}}$. The traces are offset vertically for clarity and globally fitted to our model (solid lines), the only fit parameter being the total attenuation of the drive line (input 1 in Fig. 1). For each input power $P_{\mathrm{in}}$, we note the corresponding $|g⟩\leftrightarrow |f⟩$ drive rate ${\mathrm{\Omega }}_{\mathrm{gf}}$, relative to the decay rate ${\mathrm{\Gamma }}_{ge}$.

Figure 3

Photon-photon correlations. Time-resolved power cross-correlation $g_{fe}^{(2)}(\tau )$ between modes $\widehat{f}$ and $\widehat{e}$, taken at the indicated, normalized drive rates ${\mathrm{\Omega }}_{\mathrm{gf}}/{\mathrm{\Gamma }}_{ge}$ (dots). The solid lines are calculated based on the parameters of Fig. 2, taking into account the 10 MHz detection bandwidth of the used experimental setup and with no fit parameters.

Figure 4

Pulsed excitation: photon shapes and phase correlations. (a)–(c) Density plots of the normalized photon flux in each mode, $⟨f^{†}(t)f(t)⟩/{\mathrm{\Gamma }}_{ef}$ (a) and $⟨e^{†}(t)e(t)⟩/{\mathrm{\Gamma }}_{ge}$ (b), and of the real part of the normalized amplitude cross-correlation $⟨f(t)e(t)⟩/({\mathrm{\Gamma }}_{ge}{\mathrm{\Gamma }}_{ef}{)}^{1/2}$ (c), vs time, $t$ (horizontal axis), and squared amplitude of the excitation pulse, $A^2$ (left vertical axis). The right vertical axis shows the preparation angle ${\theta }_r$, calibrated using the data of panel (g). (d)–(f) Data (colored dots) of (a)–(c) extracted along the dashed lines. Solid lines are calculations based on the measured parameters of the transmon, the preparation angle ${\theta }_r$, and the measured bandwidth of each amplification chain. In (d),(e) we have also plotted a reference trace from the other panel (dashed lines). (g) Integrated photon fluxes, $\int ⟨f^{†}f⟩dt$ and $\int ⟨e^{†}e⟩dt$, vs $A^2$. The solid lines are a fit to the theory model, serving as a calibration for the preparation angle ${\theta }_r$. (h) Integrated amplitude cross correlation, $\int ⟨fe⟩dt$, vs $A^2$: data (dots) and corresponding theory (solid line) for the same parameters as in (g).

Figure 5

Joint quantum state tomography of the modes $\widehat{e}$ and $\widehat{f}$. (a),(b) Reconstructed moments of the temporally matched modes $\widehat{f}$ and $\widehat{e}$ (absolute values) up to fourth order for the preparation angles (a) ${\theta }_r\approx \pi$ and (b) ${\theta }_r\approx \pi /2$ (color bars). The moments are referenced to the output of the switch. The expected moments for the states $|11⟩$ and $(|00⟩+|11⟩)/\sqrt{2}$ are shown as wire frames in (a) and (b), respectively. (c) Real part of the most likely density matrix corresponding to the measured moments in (b), displaying a 91% fidelity to the state $(|00⟩+|11⟩)/\sqrt{2}$ (wire frames) and a negativity of $-0.43$. All entries of the imaginary part (not shown) have absolute value below 0.01.