Smoothed quantum-classical states in time-irreversible hybrid dynamics

We consider a quantum system continuously monitored in time which in turn is coupled to an arbitrary dissipative classical system (diagonal reduced density matrix). The quantum and classical dynamics can modify each other, being described by an arbitrary time-irreversible hybrid Lindblad equation. Given a measurement trajectory, a conditional bipartite stochastic state can be inferred by taking into account all previous recording information (filtering). Here, we demonstrate that the joint quantum-classical state can also be inferred by taking into account both past and future measurement results (smoothing). The smoothed hybrid state is estimated without involving information from unobserved measurement channels. Its average over recording realizations recovers the joint time-irreversible behavior. As an application we consider a fluorescent system monitored by an inefficient photon detector. This feature is taken into account through a fictitious classical two-level system. The average purity of the smoothed quantum state increases over that of the (mixed) state obtained from the standard quantum jump approach.

Figure 1

Scheme levels corresponding to the evolution (31). The quantum system is characterized by the states $|\pm \rangle ,$ while the classical one is characterized by the states $\left|d\right)$ and $\left|u\right).$ The transition rates are $\gamma \eta$ (blue full lines) and $\gamma (1-\eta )$ (red dashed lines). The quantum system is coupled to an external laser field with Rabi frequency $\mathrm{\Omega }.$

Figure 2

Realizations for filtered (dotted blue lines) and smooth (full red lines) states associated to Eq. (31). (a) Upper system population $\langle +\left|{\rho }_{t,T}^{\mathrm{st}}\right|+\rangle .$ (b) System purity $\mathrm{Tr}\left[{\left({\rho }_{t,T}^{\mathrm{st}}\right)}^2\right].$ (c) Classical population $\left(d\right|P_{t,T}^{\mathrm{st}}).$ (d) $\text{purity}(d,u)\equiv {\left(d\right|P_{t,T}^{\mathrm{st}})}^2+{\left(u\right|P_{t,t}^{\mathrm{st}})}^2.$ The filtered states correspond to $T=t.$ In all cases, the parameters are $\mathrm{\Omega }/\gamma =1$ and $\eta =0.8.$ For the smoothed realizations, $T$ is chosen such that $\gamma (T-t)=30.$ The vertical dashed lines are the times of the undetected events.

Figure 3

Purities of the smoothed and filtered partial states when averaged over $5\times {10}^3$ realizations (see Fig. 2). (a) and (b) correspond to $\overleftrightarrow{\mathrm{Tr}\left[{\left({\rho }_{t,T}^{\mathrm{st}}\right)}^2\right]}$ (smoothed) and $\overleftarrow{\mathrm{Tr}\left[{\left({\rho }_t^{\mathrm{st}}\right)}^2\right]}$ (filtered). (c) and (d) correspond to $\overleftrightarrow{{\left(d\right|{\rho }_{t,T}^{\mathrm{st}})}^2+{\left(u\right|{\rho }_{t,T}^{\mathrm{st}})}^2}$ (smoothed) and $\overleftarrow{{\left(d\right|{\rho }_t^{\mathrm{st}})}^2+{\left(u\right|{\rho }_t^{\mathrm{st}})}^2}$ (filtered), both objects being denoted as $\overline{\text{purity}(d,u)}.$ The parameters are $\mathrm{\Omega }/\gamma =1,$ while $\eta$ is indicated in each plot.

Figure 4

Analytical solutions and average over realizations of the filtered and smoothed states. (a) Upper system population, $\langle +\left|{\rho }_t\right|+\rangle$ analytical solution, jointly with the smoothed $\overleftrightarrow{\langle +\left|{\rho }_{t,T}^{\mathrm{st}}\right|+\rangle }$ and filtered $\overleftarrow{\langle +\left|{\rho }_t^{\mathrm{st}}\right|+\rangle }$ averaged populations. (b) Classical population, $\left(d\right|P_t)$ analytical solution, jointly with the smoothed $\overleftrightarrow{\left(d{P}_{t,T}^{\mathrm{st}}\right)}$ and filtered $\overleftarrow{\left(d\right|P_t^{\mathrm{st}})}$ averaged populations. In all cases the averages were performed with $5\times {10}^5$ realizations. The parameters are $\mathrm{\Omega }/\gamma =1$ and $\eta =0.8.$