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
Reports do not certify correctness or automatically change the problem's status. Progress policy.
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.