Generating Entangled Microwave Radiation Over Two Transmission Lines

Using a superconducting circuit, the Josephson mixer, we demonstrate the first experimental realization of spatially separated two-mode squeezed states of microwave light. Driven by a pump tone, a first Josephson mixer generates, out of quantum vacuum, a pair of entangled fields at different frequencies on separate transmission lines. A second mixer, driven by a $\pi$-phase shifted copy of the first pump tone, recombines and disentangles the two fields. The resulting output noise level is measured to be lower than for the vacuum state at the input of the second mixer, an unambiguous proof of entanglement. Moreover, the output noise level provides a direct, quantitative measure of entanglement, leading here to the demonstration of $6\mathrm{Mebit}·{\mathrm{s}}^{-1}$ (mega entangled bits per second) generated by the first mixer.

Figure 1

Principle of the experiment. (a) When pumped with a microwave tone at frequency $f_P=f_a+f_b$, a Josephson mixer (black diamond) transforms incoming quantum vacuum noise (left) on modes $a$ and $b$ into an EPR state (right). The field states are represented by their standard deviation contours in the single-mode phase space of $a$ (top), $b$ (bottom), and in the bipartite phase space (middle) spanned by $\mathbf{(}\mathrm{Re}(a),\mathrm{Re}(b)\mathbf{)}$ and $\mathbf{(}\mathrm{Im}(a),\mathrm{Im}(b)\mathbf{)}$, where $\mathrm{Re}(a)=(a+a^{†})/2$ and $\mathrm{Im}(a)=(a-a^{†})/2i$ are the in-phase and out-of-phase quadratures of mode $a$. In each plot, a solid black circle sets the scale of vacuum noise and a dashed circle sets the scale of amplified vacuum noise. Quantum entanglement in the output fields is observed in the bipartite phase space where cross correlations go beyond quantum uncertainty by a squeezing factor $e^{-r}$. (b) In order to demonstrate entanglement at the output of this mixer named “entangler,” a second, identical mixer named “analyzer” is placed in series and pumped by the same tone with phase difference $\Delta \phi$. Entanglement at the input of the analyzer is revealed by measuring, at the analyzer output (dashed frame), a lower noise level on mode $a$ or $b$ than for amplified vacuum (dashed circle) for a given phase $\Delta \phi$. The smallest (largest) output noise level occurs for opposite (equal) squeezing factor at $\Delta \phi =\pi (0)$, and can ideally go as low as the vacuum noise level.

Figure 2

Schematics of the experimental setup. Each Josephson mixer consists of a ring of Josephson junctions coupling two $\lambda /2$ superconducting resonators addressed via a 180° hybrid coupler or a single ended port. Both mixers are designed with the same geometry as in Ref. 12, and their resonance frequencies are matched at $f_a=5.578\mathrm{GHz}$ and $f_b=8.812\mathrm{GHz}$ using two independent flux biases. The pump frequencies are set to $f_P=14.390\mathrm{GHz}$. Microwave circulators separate the input and output of the entangler and analyzer. Input ports are represented as open circles, and at each output port the double triangle symbolizes the low-noise amplifying measurement setup with total gain $G_{\mathrm{LNA}}$.

Figure 3

(a) Protocol of the scattering coefficient measurements by a vector network analyzer connected between the $a,b$ input and the $a,b$ output ports. The setup is calibrated by turning on and off each Josephson mixer separately. Losses are modeled as field splitters of transparency ${\alpha }^2$ and ${\beta }^2$ coupling a cold load to the signals. (b) Color traces: Transmission measurements of $|t_{a\to a}{|}^2$ as a function of phase difference $\Delta \phi$ between both pump signals. The gain of the analyzer is set to $G_A={cosh}^2{r}_A=10$ (solid gray line). Each trace and color corresponds to a different gain for the entangler $G_E={cosh}^2{r}_E=0.2,0.8,1.8,3.2,5,7.2,9.8\mathrm{dB}$. Dashed lines are fits to the data using Eq. (4) and the single fit parameter $\overline{\beta }/\overline{\alpha }=0.945$. Together with an independent, in situ noise calibration, this value leads to $\alpha =0.33\pm 0.05$ and $\beta =0.36\pm 0.05$ [14].

Figure 4

(a) Color traces: Variance of the output mode $(\Delta a_{\mathrm{out}A}{)}^2$ referred to the case of vacuum input on the analyzer [divided by $\cosh \left(2r_A/2\right)$] as a function of phase difference $\Delta \phi$, determined by measuring the spectral density of the noise at the analyzer a output when only quantum noise enters the entangler $a,b$ inputs. An absolute calibration allows exact conversion between both quantities with an error of at most $\pm 2.5\%$ [14]. Each color corresponds to the same gain of the entangler $G_E$ as in Fig. 3 with a fixed gain on the analyzer $G_A={cosh}^2{r}_A=10$. The horizontal line at ${\sigma }^2=1$ represents the measured noise for amplified vacuum at the output of the analyzer ($r_E=0$). For $\Delta \phi$ close to $\pi$, the measured noise goes below this level, an evidence of entanglement. Dashed lines: Predicted variance using Eq. (5) extended to the unbalanced loss case using $\alpha =0.37$ and $\beta =0.40$. (b) Dots: Noise level measured at $\Delta \phi =0$ (antisqueezing) and $\Delta \phi =\pi$ (squeezing) as a function of gain $G_E$ for $G_A=10$. The size of the dots is larger than the error bar. Solid lines: Prediction using Eq. (6), extended to unbalanced losses as in (a). Colored area: Consistent values of the noise using the uncertainty in the calibration of the losses $\alpha$ and $\beta$ [14]. Dashed lines: Same prediction but without losses, $\alpha =\beta =0$. (c) Solid dots: Logarithmic negativity measure of entanglement, with errors bars. Solid squares: Entanglement purity. Lines: Theoretical predictions using the parameters of (b).