Quantum Sensing of Birefringence Beyond the Classical Limit with a Hyper-Entangled SU(1,1) Interferometer

Abstract

Quantum interferometric sensing plays a crucial role in a wide range of applications, including quantum metrology, quantum imaging, and quantum lithography, where minute phase shifts carry valuable physical information. The strength of quantum sensing lies in surpassing classical sensitivity limits, particularly through the use of quantum entanglement and squeezing to suppress optical shot noise. Birefringence sensing is crucial for various applications, as it provides detailed information about the material's structure, stress, composition, and environmental conditions. We present an interferometric scheme for detecting unknown small birefringence beyond the shot-noise limit of sensitivity that leverages the hyper-entanglement within a pair of polarized nonlinear SU(1,1) interferometers, coupled by the birefringence. Specifically, two pairs of crossed-polarization nonlinear media, both generate and measure two-mode quantum light that is squeezed and polarization-entangled. We present a complete theoretical analysis of the interferometer's sensitivity to small birefringence under realistic conditions of gain and internal loss, illuminating the potential for enhancement of the sensitivity by 3-15dB in practical, real-world experiments (the exact achievable enhancement is governed solely by the loss).

Figures (6)

• Figure 1: Schematic layout of the proposed dual SU(1,1) interferometer for measuring birefringence. Two pairs of OPAs are arranged in series (one for each polarization, $H$ and $V$) and the target birefringent sample in between. Quarter/half-wave plates are incorporated on the intermediate beam paths to manipulate the polarization of the pump (the detection basis) and the signal+idler (the polarization Bell state). The birefringent phase of the sample is $\varphi$, and its axis is at an angle $\delta$ with respect to the crystal axes of the OPAs. The vertical dashed lines (marked (0)...(4)) denote various intermediate planes in the configuration where the polarization state can be calculated. The figure shows the symmetric configuration, where the target birefringent sample is placed between two quarter-wave plates (at the same angle), but one can equally place the sample also before both wave-plates or after them. Our model can calculate sensitivity maps for all three cases (and all four Bell states.)
• Figure 2: Sensitivity vs parametric gain (a) and internal losses (b). Both graphs show the relative sensitivity squared $S^2=\left(\delta\varphi/\delta\varphi_c\right)^2$ as a function of the birefringent phase $\varphi$ for (a) varying squeezing gain ($g=1,1.5,2$) and (b) varying internal loss ($l=0\%, 20\%,40\%$). When probing the sample at an angle $\delta = \pi/2$ with $| \Phi^+ \rangle =\frac{1}{\sqrt{2}}(|HH \rangle+|VV\rangle)$ state and measuring the total number of idler horizontal ($iH$) intensity and seed at signal horizontal ($sH$), we observe enhanced sensitivity around $\varphi=0$. Sensitivity improvements of -5.44 dB for $g=1$, -9.98 dB for $g=1.5$, and -14.36 dB for $g=2$ are observed.
• Figure 3: Representative sensitivity maps for the different Bell states: We present the relative sensitivity $S^2$ when the birefringent sample is placed symmetrically between the $\lambda/4$ plates, which are set in accordance with each one of the Bell states (according to table \ref{['vis']}) and for different seeding and detection polarizations measurement in $H$--$V$ basis: The columns represent the four Bell states and the rows represent seeding -- detection configurations, as follows - In the top (bottom) row we seed the signal at horizontal (vertical) polarization and detect the idler intensity at the same polarization. One can observe that for $| \Phi^+ \rangle =\frac{1}{\sqrt{2}}(|HH \rangle+| VV \rangle)$(a & b), both polarizations show the same map (as expected from the symmetry of $| \Phi^+\rangle$) with quantum enhanced sensitivity along the $\varphi_0\!=\!0$ line, but excluding regions at $\delta=\pi/4, 3\pi/4$. For $| \Phi^- \rangle =\frac{1}{\sqrt{2}}(|HH \rangle-| VV \rangle)$(c & d), the two polarizations show enhanced sensitivity along opposite lines; $\varphi_0=0$ for $H$ detection (excluding $\delta=\pi/4, 3\pi/4$), and $\varphi_0 = \pm\pi$ for $V$ detection (independent of the sample orientation $\delta$). For the $| \Psi^+ \rangle =\frac{1}{\sqrt{2}}(|HV \rangle+| VH \rangle)$ state (e & f), show no improvement in sensitivity for the $H$ detection, whereas $V$ detection shows a surprising "wiggly" striped pattern of improved sensitivity. For the singlet $| \Psi^- \rangle =\frac{1}{\sqrt{2}}(|HV \rangle-| VH \rangle)$ state (g & h), shows the same behavior as $| \Phi^+ \rangle =\frac{1}{\sqrt{2}}(|HH \rangle+| VV \rangle)$ state. Internal losses of $10\%$ in intensity are assumed for all the maps.
• Figure 4: Sensitivity maps for the different Bell states (shifted target location): We present the relative sensitivity $S^2$ when the birefringent sample is placed before (or after) the two $\lambda/4$ plates (dashed plane 1 (or 2) in configuration presented in main article), which are set in accordance with each one of the Bell states (according to table given in main article) and for different seeding and detection polarizations in $H$-$V$ basis: The columns represent the four Bell states and the rows are two seeding - detection configurations - In the top (bottom) row we seed the signal at horizontal (vertical) polarization and detect the idler intensity at the same polarization. One can observe that for $| \Phi^+ \rangle =\frac{1}{\sqrt{2}}(|HH \rangle+ |VV \rangle)$(a & b), both polarizations show the same map (expected from the symmetry of $| \Phi^+\rangle$) with quantum enhanced sensitivity along the $\varphi_0\!=\!0$ line (excluding $\delta=\pi/4, 3\pi/4$). For $| \Phi^- \rangle =\frac{1}{\sqrt{2}}(|HH \rangle- | VV \rangle)$(c & d), the two polarizations show enhanced sensitivity along opposite lines: at $\varphi_0=0$ for $H$ detection (excluding $\delta=\pi/4, 3\pi/4$), and $\varphi_0 = \pm \pi$ for $V$ detection at $\delta=0,\pi/2,\pi$ (in form of isolated minute regions). For the $| \Psi^+ \rangle =\frac{1}{\sqrt{2}}(|HV\rangle+ | VH \rangle)$ state (e & f), $H$ detection do not show any improvement in sensitivity, whereas $V$ detection show enhanced sensitivity in isolated minute regions around $\delta=\pi/8, 5\pi/8$. For the singlet $| \Psi^- \rangle =\frac{1}{\sqrt{2}}(|HV\rangle- | VH \rangle )$ state (g & h), both polarizations show the same behavior as $| \Phi^+ \rangle =\frac{1}{\sqrt{2}}(|HH \rangle+ | VV \rangle)$, but shifted by $\delta=\pi/8$. Internal losses of $10\%$ in intensity are assumed for all the maps.
• Figure 5: Sensitivity maps for the different Bell states (different polarization for detection vs. seeding): We present the relative sensitivity $S^2$ for the same configuration of figure \ref{['biref_qwp_qwp_plots']} above, but for detecting at the opposite polarization relative to the seed in $H$-$V$ basis (seed at signal-$H$ and detection at idler-$V$, or vice versa). One can observe that now both $| \Phi^\pm \rangle =\frac{1}{\sqrt{2}}(|HH \rangle\pm | VV \rangle)$(a & b, c & d) show no improvement in sensitivity (though interesting structure). For the $| \Psi^+ \rangle =\frac{1}{\sqrt{2}}(|HV\rangle+ | VH \rangle )$ state (e & f), the map is the same as in figure \ref{['biref_qwp_qwp_plots']}, but shifted in $\delta$ by $\pi/4$. For the singlet $| \Psi^- \rangle =\frac{1}{\sqrt{2}}(|HV\rangle- | VH \rangle )$ state (g & h), both polarizations show improved sensitivity at $\varphi_0=\pm \pi$, independent of the sample orientation $\delta$.