Estimation of the covariance matrix of macroscopic quantum states

For systems analogous to a linear harmonic oscillator, the simplest way to characterize the state is by a covariance matrix containing the symmetrically ordered moments of operators analogous to position and momentum. We show that using Stokes-like detectors without direct access to either position or momentum, the estimation of the covariance matrix of a macroscopic signal is still possible using interference with a classical noisy and low-intensity reference. Such a detection technique will allow one to estimate macroscopic quantum states of electromagnetic radiation without a coherent high-intensity local oscillator. It can be directly applied to estimate the covariance matrix of macroscopically bright squeezed states of light.

Figure 1

Schematic figure of the Stokes-like measurement with an uncorrelated classical, noisy, and low-intensity reference interfering with the macroscopically bright signal. T stands for the transmittance of a beam splitter, $\phi$ is a phase shift between the signal and the reference, and $I_1$ and $I_2$ are photocurrents from standard intensity detectors.

Figure 2

The used parametrization of Gaussian states. One can get an arbitrary Gaussian state from a thermal state (with a proper standard deviation $r\ge 1$) using a squeezing and displacement. The shape of the Gaussian state is defined by squeezing: by using a quadrature squeezing in direction $\alpha$ with a magnitude of $q\ge 1$, we get the ellipse of the covariance matrix. [Note that $b=rq$ and $c=r/q$ are the square roots of the eigenvalues of the covariance matrix ($b\ge c$) and $\alpha$ defines its eigendirection.] The displacement of the ellipse is given by its direction $\beta$ and its magnitude $d$. [Note that this corresponds to the mean of the Gaussian state: $\langle x\rangle =dcos\beta ,\langle p\rangle =dsin\beta )$.]

Figure 3

MSE of the state estimation of a general signal as a function of the displacement ratio (left figure) and NER (right figure) of the reference. Cyan (light) lines correspond to the estimation of parameter $b$, orange (medium) to parameter $c$, and purple (dark) to parameter $d$. In the left figure we used $\mathrm{\Delta }=1$. In the right figure we used both a displaced thermal reference (dashed lines, $\gamma =1$) and displaced$+$squeezed thermal reference (dotted lines, $\gamma =0.5$). The MSE is calculated using $N={10}^5$ Gaussian states and signal parameters: $b_S=237,c_S=86,{\alpha }_S=0.7,d_S=158,{\beta }_S=0.2$. Note that the figures look qualitatively the same for different parameters.

Figure 4

Estimates of signal parameters as a function of the number of measurements. In the left figure the cyan (light) lines correspond to the estimates of parameter $b$, orange (medium) to parameter $c$, and purple (dark) to parameter $d$. In the right figure the red (dark) lines correspond to the estimates of parameter $\alpha$ and green (light) to parameter $\beta$. We used both displaced thermal reference (dashed lines, $\gamma =1$) and displaced$+$squeezed thermal reference (dotted lines, $\gamma =0.5$) and compared the estimates with the real values of the parameter (solid lines). The estimates are calculated using $\mathrm{\Delta }=10$ and signal parameters: $b_S=237,c_S=86,{\alpha }_S=0.7,d_S=158,{\beta }_S=0.2$.

Figure 5

Different variants of the signal state with the same second moments; therefore they are indistinguishable by using only a squeezed reference.

Figure 6

MSE of the state estimation of a nondisplaced squeezed signal (left figure) and a symmetrical displaced signal (right figure) as a function of the NER of the reference. Cyan (light) lines correspond to the estimation of parameter $b$, orange (medium) to parameter $c$, and purple (dark) to parameter $d$. We used a displaced thermal reference (dashed lines, $\gamma =1$), displaced$+$squeezed thermal reference (dotted lines, $\gamma =0.5$), and classical squeezed thermal reference (solid lines, $\gamma =0$). The MSE is calculated using $N={10}^5$ Gaussian states. For the squeezed state we have $b_S=237,c_S=86,{\alpha }_S=0.7,d_S=0$, and for the displaced state we have $b_S=c_S=86,d_S=158,{\beta }_S=0.2$.