Quantum filtering of a thermal master equation with a purified reservoir

We consider a system subject to a quantum optical master equation at finite temperature and study a class of conditional dynamics obtained by monitoring its totally or partially purified environment. More specifically, drawing from the notion that the thermal state of the environment may be regarded as the local state of a lossy and noisy two-mode squeezed state, we consider conditional dynamics (“unravellings”) resulting from the homodyne detection of the two modes of such a state. Thus, we identify a class of unravellings parametrized by the loss rate suffered by the environmental two-mode state, which interpolate between direct detection of the environmental mode alone (occurring for total loss, whereby no correlation between the two environmental modes is left) and full access to the purification of the bath (occurring when no loss is acting and the two-mode state of the environment is pure). We hence show that, while direct detection of the bath is not able to reach the maximal steady-state squeezing allowed by general-dyne unravellings, such optimal values can be obtained when a fully purified bath is accessible. More generally we show that, within our framework, any degree of access to the bath purification improves the performance of filtering protocols in terms of achievable squeezing and entanglement.

Figure 1

Depiction of the configurations considered in the paper. Figure 1 describes the case of total loss $\left(\gamma =0\right)$, where only the environmental mode interacting with the system, initially in a thermal state, is measured after the interaction. Figure 1 describes the cases with purified or partially purified environment $\left(\gamma >0\right)$, where one measures both the environmental mode interacting with the system and an additional mode entangled with it (in the case $\gamma =1$ the two modes are in a global pure state). Access to a purification of the environment allows for larger degrees of asymptotic squeezing.

Figure 2

Threshold value for the purification parameter ${\gamma }_{\text{th}}\left(N\right)$ needed to observe quantum squeezing, as a function of the number of thermal excitations $N$.

Figure 3

Minimum variance $V_{x,\text{min}}$ achievable for a fixed value of the purification parameter $\gamma$. Inset: Optimal value of the thermal excitations number $N_{\text{opt}}$ minimizing the variance $V_x$ for a given value of $\gamma$.