Optimal output monitoring for parametric-oscillator squeezing
- Field
- Topic
Problem
Can any causal output measurement improve the mean conditional position squeezing achieved by ideal homodyne detection? Consider one mode with \([q,p]=i\), \(a=(q+ip)/\sqrt2\), and \(0\leq\chi<1/2\). The oscillator interacts with a vacuum Markov bath at unit damping rate. Its dynamics are Eq. (1).
The initial oscillator state is the unconditional stationary state. The measurement \(\mathsf M\) may be adaptive or non-Gaussian and may access the output field up to the current time. It does not apply control operations to the oscillator. Let \(\rho_c(t)\) be the oscillator state conditioned on the measurement record. The mean is over all records, without postselection. Define the optimum by Eq. (2).
Does \(s_{\mathrm{all}}(\chi)=1-2\chi\) hold throughout the stated range?
Source
This is a formulation for a fixed-quadrature mean cost in the parametric oscillator of Genoni, Lami, and Serafini, Sec. 6.1 [GLS16]. The extension of its monitoring optimization to arbitrary non-Gaussian output measurements is editorial.
Progress
At unit damping and zero temperature, Sec. 6.1, Eqs. (77)–(78) of the arXiv PDF, give the stationary unconditional and ideal-homodyne covariances in Eq. (3) [GLS16]. Choose the homodyne phase whose output signal is \(a+a^\dagger=\sqrt2q\). The covariance convention is \(V_{jk}=\langle\{R_j-\langle R_j\rangle,R_k-\langle R_k\rangle\}\rangle\) for \(R=(q,p)^{\mathsf T}\).
\begin{equation} \begin{aligned} V_{\mathrm{unc}}&=\operatorname{diag}\!\left(\frac1{1+2\chi},\frac1{1-2\chi}\right),\\ V_{\mathrm{hom}}&=\operatorname{diag}\!\left(1-2\chi,\frac1{1-2\chi}\right). \end{aligned} \tag{3} \end{equation}A direct ensemble argument proves optimality even for non-Gaussian monitoring. Write \(V_{q,c}=2\operatorname{Var}_{\rho_c}q\) and \(V_{p,c}=2\operatorname{Var}_{\rho_c}p\). Robertson uncertainty gives \(V_{q,c}V_{p,c}\geq1\). Jensen’s inequality and the law of total variance then imply Eq. (4) at every time.
\begin{equation} \mathbb E[V_{q,c}]\geq\mathbb E[1/V_{p,c}]\geq\frac1{\mathbb E[V_{p,c}]}\geq\frac1{(V_{\mathrm{unc}})_{pp}}=1-2\chi. \tag{4} \end{equation}The unconditional state stays stationary because measurements act only on the output. Combining Eq. (4) with the homodyne value in Eq. (3) proves \(s_{\mathrm{all}}(\chi)=1-2\chi\). This last argument is supplied here; it is not attributed to the paper.
Comment
The fixed-quadrature mean cost is completely resolved by the published homodyne solution and the direct lower-bound argument above. The latter is not a separately peer-reviewed theorem. Outcome-dependent quadrature choices, postselected costs, and feedback control on the oscillator define different optimization problems.