Quantum State Tomography of an Itinerant Squeezed Microwave Field

We perform state tomography of an itinerant squeezed state of the microwave field prepared by a Josephson parametric amplifier (JPA). We use a second JPA as a preamplifier to improve the quantum efficiency of the field quadrature measurement from 2% to $36\%\pm 4\%$. Without correcting for the detection inefficiency we observe a minimum quadrature variance which is ${68}_{-7}^{+9}\%$ of the variance of the vacuum. We reconstruct the state’s density matrix by a maximum likelihood method and infer that the squeezed state has a minimum variance less than 40% of the vacuum, with uncertainty mostly caused by calibration systematics.

Figure 1

Principle of the experiment. (a) (Left) The squeezer (SQ, in red) prepares a squeezed state whose quadrature distributions are measured for different phases $\theta$ with an efficiency $\eta$. (Right) Simulated measurement results for 20 000 realizations of creating the squeezed state and measuring it at a single $\theta$. The top graph shows the measured quadrature value versus realization number. The bottom plot is a histogram (blue circles) and Gaussian probability distribution $\mathrm{pr}(X_{\theta })$ (red curve) of this random process. (b) Graphical interpretation. The probability distribution $\mathrm{pr}(X_{\theta })$ is simply the projection of the Wigner function.

Figure 2

(a) To implement a SQ or AMP at microwaves, three microwave components are required: (i) a JPA, (ii) a directional coupler, and (iii) a circulator. Taking port (1) of the directional coupler as reference, (2) is the weakly coupled port, (3) the isolated port, and (4) the direct port. Port (2) is used to pump the JPA. Port (3) is used to apply a cancellation tone (adjusted with a room temperature attenuator and phase shifter) that nulls the pump and displaces the output of the JPA back to the origin of the phase space. (b) Schematic of the experiment. In this figure, all the microwave components and cables are considered lossless; their imperfections are absorbed into the experimentally determined total transmissivities $\xi$, $\alpha$, and $\beta$.

Figure 3

(a) Density plot of number of occurrences in a $1\mu \mathrm{V}$ bin size of the amplified quadrature voltage $V_{\theta }$ versus $\theta /2\pi$, with the SQ pump off (top) and on (bottom). (b) In particular, histograms of $V_{\theta }$ at the maximum of squeezing: data (○) and Gaussian fit (continuous lines) for the SQ pump off (blue) and on (red). (c) Noise variance $\Delta X_{\theta }^2$ in quanta units on a log scale versus $\theta /2\pi$ for the SQ pump on (red) and off (blue). The (black) line indicates our estimate of the vacuum noise level under the “best-guess” calibration.

Figure 4

Mean of 35 reconstructions of the Wigner function of the state exiting the SQ, inferred by maximum likelihood under the “best-guess” assumption, in quanta units. The faint pattern of ripples extending from the origin is caused by truncation at 30 photons of the density matrix used to represent the state. The white circle at the origin shows the full width at half maximum of the vacuum state.