Coherence-based measurement of non-Markovian dynamics in an open quantum system

We present a theoretical scheme and its experimental proof of principle for an open quantum system undergoing Markovian and non-Markovian evolutions. We exhibit these two regimes by diagnosing them with the relative entropy of coherence of two polarization qubits playing the roles of system and ancilla. These are initially prepared in a polarization maximally entangled state of a photon pair produced by spontaneous parametric down-conversion. We induce Markovian and non-Markovian regimes in the system's dynamics with the help of two auxiliary qubits, experimentally implemented by optical paths in a layout of Sagnac and Mach-Zehnder interferometers. We replicate system-environment interactions by means of an amplitude damping channel and a suitably designed inversion of it. In our scheme, one needs only two experimentally accessible parameters to achieve Markovian and non-Markovian regimes.

Figure 1

Diagram of the experiment. (a) SPDC source and $a$-photon tomography. (b) Implementation of the two-step channel: a SI-based AD channel (first step) and a MZI-based A-AD channel (second step). After the angular rotation ${\theta }_1\left({\eta }_1\right)$ made by $U_{\mathrm{HWP}}$, the undamped (damped) light exits the SI through its bottom (top) output mode ${|0\rangle }_e$ (${|1\rangle }_e$). The angular rotation ${\theta }_2\left({\eta }_2\right)$ made by $U_{\mathrm{HWP}}^{\prime }$ sends the damped light back to ${|0\rangle }_e$. A tilting glass G provides the relative phase ${\delta }_e=\pm \pi /2$, which is required for interference at the last beam splitter. Another G and a thick compensation glass (CG) are used to compensate relative time delays between ${|0\rangle }_e$ and ${|H\rangle }_s$ after ${\theta }_2\left({\eta }_2\right)$, while ${|V\rangle }_s$ is filtered out from ancilla-system coincidences in the nested MZI.

Figure 2

Theoretically calculated $C_{\text{rec}}\left[{\rho }_{a,s}\left(t_2\right)\right]$ for both the damping angle ${\theta }_1$ and the antidamping angle ${\theta }_2$ varying in the range $[0,\pi /4]$.

Figure 3

Measured density matrix of the initial state ${\rho }_{(a,s)}^{\left(\text{in}\right)}$. The targeted state is a pure, maximally entangled one: ${|\psi \rangle }_{(a,s)}\langle \psi |$. Measured purity is better than $92\%$.

Figure 4

Experimental results of $C_{\text{rec}}\left({\rho }_{a,s}\right)$ at the first step of the evolution. We exhibited coherence loss by varying ${\theta }_1$ so as to scan $C_{\text{rec}}\left({\rho }_{a,s}\right)$ over a wide range of values between 1 and zero. Error bars come from the standard deviation over ten measurements.

Figure 5

Experimental results of $C_{\text{rec}}\left({\rho }_{a,s}\right)$ and ${\mathcal{N}}_{\text{rec}}\left({\rho }_{a,s}\left(t_2\right)\right)$. We scanned ${\theta }_2$ while the damping parameter was fixed to ${\theta }_1({\eta }_1=1)=\pi /4$ and ${\theta }_1({\eta }_1=1/2)=\pi /8$ in the dashed and solid curves, respectively. Error bars come from the standard deviation over ten measurements.