Generating and verifying entangled itinerant microwave fields with efficient and independent measurements

By combining a squeezed propagating microwave field and an unsqueezed vacuum field on a hybrid (microwave beam splitter), we generate entanglement between the two output modes. We verify that we have generated entangled states by making independent and efficient single-quadrature measurements of the two output modes. We observe the entanglement witness $E_{\mathrm{W}}=-0.{263}_{-0.036}^{+0.001}$ and the negativity $N=0.{0824}_{-0.0004}^{+0.01}$ with measurement efficiencies at least $26\pm 0.1\%$ and $41\pm 0.2\%$ for channels 1 and 2, respectively. These measurements show that the output two-mode state violates the separability criterion and therefore demonstrates entanglement. This shared entanglement between propagating microwaves provides an important resource for building quantum networks with superconducting microwave systems.

Figure 1

Single squeezer model of the experiment. The squeezer (SQ) prepares a squeezed state with squeezing parameter $s$, where the variance of the squeezed quadrature is $1/\left(2s\right)$. The squeezed state (red ellipse) and the unsqueezed input (green circle) are combined on a quadrature hybrid (HY) to generate entangled modes. The hybrid has a power transmission coefficient $t$ and a power coupling coefficient $1-t$. The two output modes (orange ellipses) of the hybrid propagate onto two physically separate transmission lines and are fed to the two-channel measurement apparatus to measure quadrature amplitudes $W_1\left({\theta }_1\right)$ and $W_2\left({\theta }_2\right)$. The measurement apparatus consists of two single-quadrature measurement chains (QM1 and QM2), where each QM employs a VER as the first amplifier. All sources of loss (including loss inside the SQ) and measurement inefficiencies are modeled by introducing two fictitious beam splitters with power transmission coefficients ${\eta }_1$ and ${\eta }_2$, respectively. The two squeezed states arrive at the two VERs with fixed but uncontrolled phase shifts. We mathematically adjust the reference phases to align the squeezed states with $X_1$ and $X_2$ as illustrated.

Figure 2

Separate and joint variances. Shown are intensity plots of the measured variances of (a) $W_1({\theta }_1,{\theta }_2)$, of (b) $W_2({\theta }_1,{\theta }_2)$, and of (c) $\frac{1}{2}[W_1({\theta }_1,{\theta }_2)+W_2({\theta }_1,{\theta }_2)]$ calibrated in units of the vacuum vs the two quadrature phases ${\theta }_1$ and ${\theta }_2$. (d) An expectation of (c) predicted by the single squeezer model represented by Fig. 1. (e) The variances along the corresponding annotated lines in (c) and (d) are plotted vs quadrature phase ${\theta }_1$ of channel 1. The red squares are the sum of the green-circle line and the magenta-diamond line, where the green-circle line is shifted by $\frac{\pi }{2}$ in ${\theta }_1$. The arrow indicates the observed value of ${\Delta }_{\mathrm{EPR}}<1$.

Figure 3

Covariance matrix of the two-mode state. (a) The covariance matrix calculated from the single squeezer model parameter extracted by joint fitting of measured variances. (b) The 10 independent elements of the covariance matrix are shown for the same data as (a).

Figure 4

Thermal sweep experiment. The measured single-quadrature variances of the thermal sweep experiment (blue circles) and fit to Eq. (A1) (red solid line) for channel 1 (a) and channel 2 (b) are plotted. The residuals of the two fits are plotted in (c) and (d) where the error bars show the standard deviation of five independent measurements made at each temperature point.

Figure 5

Joint fit of the model equations to the measured variances. (a) The cartoon diagram represents the timing of the quadrature measurements. Each measurement is a 1-s-long time trace with each channel sampled at 10 $\mathrm{M}$ samples/s. Because the pumps of VER1 and VER2 are detuned from the SQ pump by 1 kHz and 50 kHz, respectively, the two VERs amplify the measured states at $10000$ different quadrature phase combinations every 1 ms (one record). In a 1-s-long time trace, we thus have 1000 records, or realizations at each quadrature phase pair. The variance at each quadrature phase pair is then calculated from the corresponding data points in each record. The measured variances of (b) $U_1\left({\theta }_1\right)$, of (c) $U_2\left({\theta }_2\right)$, and of (d) $1/2[U_1\left({\theta }_1\right)+U_2\left({\theta }_2\right)]$ are plotted vs time in each measurement record (blue dot) along with the joint fit of Eqs. (A2, A3, A4) to all of the measured variances (red solid line). The model and data show good agreement with each other and we observe moderate squeezing below vacuum fluctuation (0.5) in the variance of each mode [(b) and (c)], but more squeezing in the joint measurement (d).