SU(2)-in-SU(1,1) Nested Interferometer for High Sensitivity, Loss-Tolerant Quantum Metrology

We present experimental and theoretical results on a new interferometer topology that nests a SU(2) interferometer, e.g., a Mach-Zehnder or Michelson interferometer, inside a SU(1,1) interferometer, i.e., a Mach-Zehnder interferometer with parametric amplifiers in place of beam splitters. This SU(2)-in-SU(1,1) nested interferometer (SISNI) simultaneously achieves a high signal-to-noise ratio (SNR), sensitivity beyond the standard quantum limit (SQL) and tolerance to photon losses external to the interferometer, e.g., in detectors. We implement a SISNI using parametric amplification by four-wave mixing (FWM) in Rb vapor and a laser-fed Mach-Zehnder SU(2) interferometer. We observe path-length sensitivity with SNR 2.2 dB beyond the SQL at power levels (and thus SNR) 2 orders of magnitude beyond those of previous loss-tolerant interferometers. We find experimentally the optimal FWM gains and find agreement with a minimal quantum noise model for the FWM process. The results suggest ways to boost the in-practice sensitivity of high-power interferometers, e.g., gravitational wave interferometers, and may enable high-sensitivity, quantum-enhanced interferometry at wavelengths for which efficient detectors are not available.

Figure 1

Comparison of SQ-MZI and SISNI in theoretical simulations. (a),(b) Topology of SQ-MZI and SISNI, respectively. BHD, balanced homodyne detection. Insets show the various input states in the amplitude quadrature ($X_1$)—phase quadrature ($X_2$) phase space. For both interferometer types, a coherent state $|\alpha ⟩$ feeds the “bright input” port of the SU(2) interferometer, while a nonclassical state feeds the “dark input” port. (c) Signal gain $d\mathcal{X}/d\varphi$ versus local oscillator phase ${\varphi }_{\mathrm{LO}}$ for SQ-MZI (red) and SISNI (blue), for the lossless case. Both configurations have $|\alpha {|}^2=36$. (d) Shading indicates Wigner distributions for the output mode $\widehat{k}$ and ${\widehat{k}}_M$ versus signal phase $\phi$ (three equispaced $\phi$ values are shown in $X_1$, $X_2$ phase space) and versus external loss $L_e$. For larger loss values, degradation of SNR, i.e., overlap of uncertainty areas, is evident in the SQ-MZI case, but not that of SISNI. (e),(f) Quantum sensitivity advantage $A_q\equiv ⟨\delta {\phi }^2⟩/⟨\delta {\phi }^2{⟩}_{\mathrm{SQL}}$ in decibels, calculated from Eqs. (2) and (3), as a function of internal and external loss for (left) SISNI and (right) SQ-MZI.

Figure 2

Experimental demonstration of quantum enhancement of the SISNI. (a) Schematic of experiment. BS, $50/50$ beam splitter; PM, mirror mounted with piezoelectric transducer as phase modulator; DPD, differential photodetector; BL, beam block. Inset shows the double-$\mathrm{\Lambda }$ level structure of the PA process, which uses the D1 line of ${}^{85}\mathrm{Rb}$. The two pump beams (red arrows) are frequency degenerate, $\mathrm{\Delta }=1\mathrm{GHz}$ and $\delta =2\mathrm{MHz}$. ${\mathrm{PA}}_1$ and ${\mathrm{PA}}_2$ act as a source of two-mode squeezing and amplifier, respectively. (b) Noise of the MZI and SISNI. Graph shows measured noise of the output phase quadrature in a bandwidth of 100 kHz about 1.7 MHz, versus time as $\varphi$, the relative phase of the PAs, is scanned using ${\mathrm{PM}}_3$. MZI phase is locked to dark fringe. Traces show MZI SQL (black), implemented by blocking both PA pumps, MZI output amplified by ${\mathrm{PA}}_2$ (blue), implemented by blocking only ${\mathrm{PA}}_1$ pump, and SISNI when MZI locked at dark fringe as the SU(1,1) phase is scanned (red). (c) Measured noise spectra of MZI (black) and SISNI (red) outputs, acquired as in (b), but with PAs relative phase locked to noise minimum and a sinusoidal phase signal at 1.7 MHz applied via ${\mathrm{PM}}_1$. (d) Measured SNR, defined as spectral peak over white background level from spectra as in (c), versus phase-sensing light intensity ($I_{\mathrm{ps}}$) for MZI (black) and SISNI (red). Error bars show $\pm 1\sigma$ statistical variation. Lines show fits with $\mathrm{SNR}=A|\alpha {|}^2$. The QNG of ${\mathrm{PA}}_1$ and ${\mathrm{PA}}_2$ are $G_{q1}=4$ dB and $G_{q2}=6\mathrm{dB}$, respectively.

Figure 3

Characterization of the trade-off between gain and FWM noise in the SISNI. Horizontal axis shows QNG of ${\mathrm{PA}}_2$, adjusted by control of the pump power. Vertical axis shows the SNR advantage of the SISNI over the MZI at equal coherent state input power, with positive values indicating advantage for the SISNI. Points and error bars indicate mean and $\pm 1$ standard deviation from repeated measurements. Red, blue, and magenta data points and curves correspond to ${\mathrm{PA}}_1$ QNG levels 4, 6, and 8 dB, respectively. Curves show fits based on a minimal PA noise model [30, 36].