Optimal circular dichroism sensing with quantum light: Multiparameter estimation approach

The measurement of circular dichroism (CD) has widely been exploited to distinguish the different enantiomers of chiral structures. It has been applied to natural materials (e.g., molecules) as well as to artificial materials (e.g., nanophotonic structures). However, especially for chiral molecules the signal level is very low and increasing the signal-to-noise ratio is of paramount importance to either shorten the necessary measurement time or to lower the minimum detectable molecule concentration. As one solution to this problem, we propose here to use quantum states of light in CD sensing to reduce the noise below the shot noise level encountered when using coherent states of light. Through a multiparameter estimation approach, we identify the ultimate quantum limit of the precision in CD sensing, allowing for general schemes including additional ancillary modes. We show that the ultimate quantum limit can be achieved by various optimal schemes. These include using a pair of Fock-state probes in a direct sensing configuration and pairs of twin beams in an ancilla-assisted sensing configuration, for both of which photon number-resolved detection is shown to be the optimal measurement. These optimal schemes offer a significant quantum enhancement even in the presence of some additional system loss. The near optimality of a scheme using a single twin beam in a direct sensing configuration is also shown for cases where the actual CD signal is very small. Alternative optimal schemes involving single-photon sources and detectors are also proposed. This work paves the way for further investigations of quantum metrological techniques in chirality sensing.

Figure 1

(a) Traditional CD sensing: TCD is experimentally obtained by measuring the intensity difference of the transmitted LCP ($I_{\text{L}}$) and RCP ($I_{\text{R}}$) upon propagation through a chiral medium. (b) Quantum CD sensing: The ancilla-assisted CD sensing scheme is modeled quantum mechanically by the two signal modes corresponding to LCP and RCP and arbitrary ancillary modes that may be entangled with the signal modes. Beam splitters with transmittances $T_{\text{L}\text{(R)}}$ and ${\eta }_{\text{L}\text{(R)}}$ express two distinct processes in each mode: $T_j$ expresses the transmittance of each polarization mode through the chiral medium, whereas ${\eta }_j$ addresses extra loss of each mode such as nonunity channel transmission and detection efficiency of a detector.

Figure 2

(a) The optimal ratio $r_{\text{opt}}$ as a function of logarithmic $x={\eta }_{\text{L}}{T}_{\text{R}}/{\eta }_{\text{R}}{T}_{\text{L}}$ for a coherent state input. The $x$ is replaced by $x={\eta }_{\text{L}}{T}_{\text{R}}(1-{\eta }_{\text{R}}{T}_{\text{R}})/{\eta }_{\text{R}}{T}_{\text{L}}(1-{\eta }_{\text{L}}{T}_{\text{L}})$ for the optimal state input achieving the UQL to the precision of CD sensing. (b) The optimal ratio $r_{\text{opt}}$ is shown as a function of logarithmic $x_{\text{L}}=T_{\text{L}}$ and $x_{\text{R}}=T_{\text{R}}$ for a coherent state input when ${\eta }_{\text{L}}={\eta }_{\text{R}}$. The axis labels are transformed to $x_{\text{L}}=T_{\text{L}}(1-{\eta }_{\text{L}}{T}_{\text{L}})$ and $x_{\text{R}}=T_{\text{R}}(1-{\eta }_{\text{R}}{T}_{\text{R}})$ for the optimal state input.

Figure 3

(a) Quantum enhancement $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{CB}}/\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{UQL}}$ in terms of transmittance $T$ for $\eta =1,0.8,0.5$ when $T_{\text{L/R}}=T$ and ${\eta }_{L/R}=\eta$ can be assumed. (b) Quantum enhancement $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{CB}}/\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{UQL}}$ in terms of $T_{\text{L}}$ and $T_{\text{R}}$ for $\eta =0.8$.

Figure 4

(a) Quantum enhancement $\text{Var}{\left(\check{\mathrm{\Gamma }}\right)}_{\text{CB}}/\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{TMSV}}$ in terms of $T_{\text{L}}$ and $T_{\text{R}}$ for balanced losses $\eta =0.8$. (b) The normalized difference between $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{TMSV}}$ and $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{UQL}}^{\text{opt}}$ for $\eta =0.8$. Here, $N_{\text{tot}}=2$ is assumed as an example.

Figure 5

(a) The normalized difference between the CR bound for PNRD and the QCR bound $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{TMSV}}$ for balanced losses $\eta =0.8$. (b)The normalized difference between the CR bound for PNRD and the UQL $\text{Var}{\left({\check{\mathrm{\Gamma }}}_{-}\right)}_{\text{UQL}}^{\text{opt}}$ for $\eta =0.8$. Here, $N_{\text{tot}}=2$ is assumed as an example.