Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer

TL;DR

This paper demonstrates a highly loss-tolerant SU(1,1) interferometer that surpasses shot noise limit by 2.3 dB, maintaining supersensitivity even with 80% detection loss, advancing quantum metrology and imaging.

Contribution

It introduces a strongly unbalanced SU(1,1) interferometer with enhanced loss tolerance, enabling sub-shot-noise phase sensitivity under high detection losses.

Findings

Achieved 2.3 dB phase sensitivity beyond shot noise limit.

Maintained supersensitivity with up to 80% detection loss.

Enabled potential applications in quantum imaging and spectral ranges with inefficient detection.

Abstract

In an unseeded SU(1,1) interferometer composed of two cascaded degenerate parametric amplifiers, with direct detection at the output, we demonstrate a phase sensitivity overcoming the shot noise limit by 2.3 dB. The interferometer is strongly unbalanced, with the parametric gain of the second amplifier exceeding the gain of the first one by a factor of 2, which makes the scheme extremely tolerant to detection losses. We show that by increasing the gain of the second amplifier, the phase supersensitivity of the interferometer can be preserved even with detection losses as high as 80%. This finding can considerably improve the state-of-the-art interferometry, enable sub-shot-noise phase sensitivity in spectral ranges with inefficient detection, and allow extension to quantum imaging.

Equations

$$\Delta \varphi =\frac{\Delta N}{\left|\frac{\partial ⟨N_f⟩}{\partial \varphi }\right|},$$

$$\Delta {\varphi }_{SNL}=\frac{1}{2\sqrt{⟨N(\Delta \lambda ,\Delta \theta )⟩}},$$

$$D=\frac{2{\lambda }_p}{\delta n_{air}},$$

$$⟨N_f⟩=\eta \nu \left[\mu |S(\varphi ){|}^2+(1-\mu ){\sinh }^2{r}_2\right],$$

$$\begin{array}[]{rcl}S(\phi)&=&\sinh r_{1}\cosh r_{2}\,e^{-i\phi}-\cosh r_{1}\sinh|r_{2}|\,e^{i\phi}\,.\end{array}$$

$$\begin{array}[]{rcl}\langle N_{f}\rangle&\simeq&\frac{1}{4}\eta\nu\mu\exp{(2(r_{1}+|r_{2}|))}\sin(\phi)^{2}\\ &+&\eta\nu(1-\mu)\exp(2|r_{2}|)/4.\end{array}$$

$$⟨N(\Delta \lambda ,\Delta \theta )⟩=\nu {\sinh }^2(r_1)\simeq \nu \exp (2r_1)/4.$$

$$\begin{array}[]{rcl}\langle N_{f}\rangle&\simeq&\eta\mu\langle N(\Delta\lambda,\Delta\theta)\rangle\exp(2|r_{2}|)\sin(\phi)^{2}\\ &&+\eta\nu(1-\mu)\exp(2|r_{2}|)/4.\end{array}$$

$$⟨N_f⟩\simeq \eta \mu ⟨N(\Delta \lambda ,\Delta \theta )⟩\exp (2|r_2|)\sin (\varphi {)}^2+B.$$

$$\begin{array}[]{ccl}r_{1}&=&\operatorname{asinh}\Big{(}\sqrt{\dfrac{\langle N(\Delta\lambda,\Delta\theta)\rangle}{\langle N_{2}(\Delta\lambda,\Delta\theta)\rangle}\sinh^{2}r_{2}}\Big{)},\\ \nu&=&\dfrac{\langle N(\Delta\lambda,\Delta\theta)\rangle}{\sinh^{2}(r_{1})}.\end{array}$$

Full text

The sensitivity of an interferometric measurement on a phase shift depends on the state of light used as a probe and the measurement scheme. A ‘standard’ precision is provided by a coherent state fed into a Mach-Zender interferometer, the so-called shot noise limit (SNL). A measurement beating this limit is said to be supersensitive. In order to make super-sensitive phase measurements, quantum resources can be used. First proposed Caves (1981) and experimentally tested Xiao et al. (1987); Grangier et al. (1987) in the 1980-s, squeezed light is now used for gravitational wave detection Abadie J. et al (2011); Aasi et al. (2013) beyond the shot noise limit. Squeezed states can improve the sensitivity in the presence of loss Gea-Banacloche and Leuchs (1987), a finite interference visibility Gea-Banacloche and Leuchs (1989) and their use is compatible with power recycling Brillet et al. (1991). Supersensitivity can also be achieved with other quantum states Holland and Burnett (1993); Lee et al. (2002), which, however, are difficult to produce.

Besides the input state and the detection scheme, one can also modify the interferometer. Yurke et al. Yurke et al. (1986) proposed to use cascaded optical parametric amplifiers (OPAs) instead of the passive beam-splitters of conventional interferometric setups, the phase sensitive response of the OPAs giving rise to interference patterns. Such interferometers, usually called SU(1,1) interferometers, can display phase super-sensitivity without seeding the amplifiers Yurke et al. (1986). Seeding can be used in order to increase the number of sensing photons and therefore the overall sensitivity Plick et al. (2010). It was recently noted theoretically that such a scheme involving two amplifiers can help overcoming the deleterious effects of optical losses Ou (2012); Marino et al. (2012) on phase sensitivity. Note that losses occurring inside the interferometer have a different impact on the phase sensitivity than losses outside of the interferometer. These two kinds of losses are therefore distinguished in the following and respectively called internal and external/detection losses. The influence of the latter kind of losses can be suppressed by the second amplifier. Indeed, amplifying the signal with the second OPA at the output of the interferometer while keeping the probing field constant eventually eliminates the effect of detection losses Sparaciari et al. (2016); Manceau et al. (2017).

Optical SU(1,1) interferometers have been implemented using two cascaded four-wave mixers Jing et al. (2011); Kong et al. (2013). Recent experiments with this interferometer operating in the continuous-wave regime with seeding have demonstrated phase super-sensitivity Hudelist et al. (2014) using homodyne detection. A truncated version, i.e. without a second amplifier, of the same scheme Anderson et al. (2016) has also demonstrated the possibility of supersensitive phase measurement. The effect of internal losses on the quantum noise of an SU(1,1) interferometer has been studied in detail in Ref. Xin et al. (2016). Finally, an atom SU(1,1) interferometer has shown sensitivity beyond the SNL Linnemann et al. (2016) and more complex schemes based on SU(1,1) interferometry such as an atom-light hybrid interferometer Chen et al. (2015) or a so called pumped-up SU(1,1) interferometer Szigeti et al. (2017) are also currently investigated.

In this letter we report phase supersensitivity in an SU(1,1) interferometer formed by two cascaded nonlinear $\chi^{(2)}$ crystals. Our goal is to overcome the negative effect of detection losses by pumping the second OPA stronger than the first one. In the previous realizations of the SU(1,1) interferometer, both OPAs had the same absolute values of the parametric gain. However as shown theoretically in Ref. Manceau et al. (2017), the gain unbalancing leads to an increased tolerance of the setup to the detection losses. The idea is illustrated in Fig. 1.

The interferometer consists of two degenerate parametric amplifiers, DOPA1 and DOPA2, the first one transforming an input vacuum state into a squeezed vacuum state, which then acquires a phase $\phi$ to be measured. Two squeezed vacuum states differing by their phases $\phi_{1}$ and $\phi_{2}=\phi_{1}+\Delta\phi$ can be distinguished whenever their Wigner functions (shown by orange and green colors) do not completely overlap. Such squeezed vacua have a higher distinguishability than two coherent states with the same photon number and the same phase difference $\Delta\phi$. The second amplifier DOPA2 applies squeezing opposite to the one of DOPA1, and the resulting output states are now squeezed in other directions, and differently as long as $\phi_{1}\neq\phi_{2}$. Importantly, the squeezing operation does not change their distinguishability. In the presence of low detection efficiency $\eta$, represented in Fig. 1 by a beamsplitter, the states change dramatically, so that the noise in the squeezed quadrature becomes almost equal to the shot noise. At the same time, the spread of the anti-squeezed quadrature is only reduced by a factor $\sqrt{\eta}$, and will remain different for the two states as long as it is initially large. Therefore, provided that the DOPA2 gain is high enough, and the states at its output are highly squeezed, they are still distinguishable after the losses are applied. Note that the smaller the detection efficiency $\eta$, the stronger squeezing in DOPA2 is required to reach a certain phase sensitivity Manceau et al. (2017).

Although the same principle of unbalancing can be applied to various configurations of an SU(1,1) interferometer (seeded or unseeded, with direct or homodyne detection) and even to an SU(2) interferometer fed with squeezed light and followed by another squeezer Manceau et al. (2017), here we use the simplest configuration of an unseeded interferometer with direct intensity detection. The robustness of the measured phase sensitivity against detection losses is tested for various parametric gains of the DOPAs. We achieve a high parametric gain by using intense picosecond pump pulses and show the possibility to overcome the detrimental effect of losses for large parametric gain of the second DOPA.

The setup is depicted in Fig. 2. The SU(1,1) interferometer consists of two cascaded $3$ mm long $\beta$-Barium Borate (BBO) crystals cut for collinear frequency-degenerate type-I phase matching. The pump is the second harmonic of a Spectra Physics Spitfire Ace system with a $5$ kHz repetition rate, $1.5$ ps pulses and a central wavelength of $\lambda_{p}=400$ nm. The beam diameter is $0.2$ mm full width at half maximum (FWHM) inside the SU(1,1) interferometer. Two dichroic mirrors are subsequently used to eliminate the pump beam. The parametric down converted (PDC) light generated by the SU(1,1) interferometer is spatially filtered by a pinhole in the focal plane of a $f=90$ cm focal length lens selecting an angular bandwidth $\Delta\theta$. An interference filter further selects a spectral bandwidth $\Delta\lambda$ around the central wavelength $\lambda_{s}=800$ nm. After the spatial and spectral filtering, the PDC light is detected using a low-noise charge integrating p-i-n diode (PD)Hansen et al. (2001); Iskhakov et al. (2009); Agafonov et al. (2009) (see supplementary).

The first crystal BBO1 is fixed on a translation stage. The distance $d$ between the two crystals can therefore be changed. Due to the dispersion of the air Pérez et al. (2014), the squeezed light and the pump acquire different phase shifts $\phi_{s}\neq\phi_{p}$ in the air gap, and the phase $\phi=\phi_{s}-\phi_{p}/2$ of the squeezed state at the input of DOPA2 (Fig. 2) can be scanned by adjusting the distance $d$. A set of $4000$ post-processed pulses is measured for each position of BBO1. To avoid high uncertainty in the measured mean photon number of the interferometer output light, the pump intensity fluctuations are eliminated through pulse post-selection (see supplementary).

By measuring the transmission of the setup at $\lambda_{s}=800$ nm, we found that the internal losses, caused by reflections on a single facet of each BBO crystal, are $3\%$. The external transmission of the setup, including the detector quantum efficiency, is measured to be $\eta=77\%$. It corresponds to the losses for a set of plane waves within the $\Delta\lambda$, $\Delta\theta$ bandwidths of the frequency and spatial filters. Assuming Gaussian noise and sufficiently small phase fluctuations, the phase sensitivity $\Delta\phi$ is given by

[TABLE]

$\Delta N$ including the intrinsic photon noise of the measured light and the detector noise and $\langle N_{f}\rangle$ being the mean number of photons at the interferometer output. For a given position $d$ of BBO1, both values are determined from the set of post-processed pulses. The phase sensitivity is then calculated according to Eq. 1.

The distance between the two crystals is converted into a phase knowing the periodicity of the interference (see supplementary). Figure 3 presents typical phase sensitivity measurements. The shot noise limited phase sensitivity for a set of plane waves within the $\Delta\lambda$, $\Delta\theta$ bandwidths, $\Delta\phi_{SNL}$, is defined as

[TABLE]

with $\langle N(\Delta\lambda,\Delta\theta)\rangle$ being the mean number of photons inside the interferometer within the $\Delta\lambda$, $\Delta\theta$ bandwidths Manceau et al. (2017). It is measured by removing BBO2 and registering the number of photons within the $\Delta\lambda$, $\Delta\theta$ bandwidths with an avalanche photodiode (APD) after attenuating the beam with neutral density filters. The value of $\langle N(\Delta\lambda,\Delta\theta)\rangle$ is found from the APD count rate knowing the APD detection efficiency $\eta_{APD}=47\%$ and the filters transmission. Note that the resulting number of photons is considerably less than the number of photons in a single Schmidt mode $N_{mode}=\sinh(r_{1})^{2}$ because of the strong spatial filtering. For a single Schmidt mode of the multimode OPA, this narrowband detection leads to a reduction of quantum efficiency Rytikov and Chekhova (2008); Agafonov et al. (2009); Pérez et al. (2015) from $\eta$ to $\eta_{SM}=\nu\eta$. Here, $\nu$ corresponds to the transmission of the spatial and frequency filters for the Schmidt mode under consideration.

The SNL phase sensitivity associated with a single Schmidt mode, $\Delta\phi_{SM}=\frac{1}{2\sinh(r_{1})}$, can still be beaten in theory with our large values of $|r_{2}|$ Sparaciari et al. (2016); Manceau et al. (2017). However, the non ideal visibility, here being $\mathcal{V}=97\%$ for the given spatial and frequency filtering (see supplementary), is a limiting factor for the best phase sensitivity achievable Gea-Banacloche and Leuchs (1989). Indeed, as the working point of this interferometer is close to the dark fringe, the non-unity visibility acts as an extra source of noise. $\Delta\phi_{SM}$ can therefore not be reached with the current experimental configuration. At the same time, as the SNL is defined for coherent light, it is still determined by the number of photons inside the interferometer within the chosen bandwidths, $\langle N(\Delta\lambda,\Delta\theta)\rangle$, regardless of the Schmidt mode structure.

Fig. 3a shows two measurements with different parametric gain values $r_{1}$ of the first crystal, and therefore different mean numbers of photons $\langle N(\Delta\lambda,\Delta\theta)\rangle$, varied by changing the phase matching. The blue points are for $\langle N(\Delta\lambda,\Delta\theta)\rangle=1.7$ photons and $r_{1}=1.5$. The corresponding shot noise limited phase sensitivity is $\Delta\phi_{SNL}=0.38$ rad (dashed blue line, the uncertainty shown by the blue rectangle). The best phase sensitivity is $2.3$ dB below the SNL. For a larger number of photons in the interferometer, $\langle N(\Delta\lambda,\Delta\theta)\rangle=11$, corresponding to $r_{1}=2.7$ and the gain of the second crystal almost the same, the SNL is overcome by $1.2$ dB (red points). The inset shows the $\langle N_{f}(\phi)\rangle$ dependences.

Fig. 3b shows that a larger parametric gain of the second crystal leads to an improved sensitivity. Indeed, for $|r_{2}|=3.9$ (blue points), the best phase sensitivity is $\Delta\phi_{min}=0.33$ rad while a larger parametric gain $|r_{2}|=5.2$ obtained by increasing the pump power gives $\Delta\phi_{min}=0.16$ rad (red points). The number $\langle N(\Delta\lambda,\Delta\theta)\rangle=4.8$ is kept constant by controlling the phase matching of BBO1. Hence the shot noise limit is the same in both cases and it is not beaten for the lower gain case.

Finally, we test the tolerance of the scheme to detection losses. For a given set ($r_{1},|r_{2}|$) of parametric gains, we vary the loss by adding neutral density filters before the detection. Figure 4 shows the measurement results for three sets of parametric gains. Theoretical dependences taking into account the detector noise and the non-ideal visibility are also shown (see supplementary). The number of photons is kept constant, $N(\Delta\lambda,\Delta\theta)=4.5\pm 0.5$, for all three cases. The parametric gain of the second crystal is reduced from $|r_{2}|=5.2$ (red points) to $|r_{2}|=4.7$ (blue points) and $|r_{2}|=4$ (green points). We can observe that for a large unbalancing of the interferometer, $|r_{2}|/r_{1}=2.5$ (red curve), the phase sensitivity is robust against the added losses, the system being still supersensitive for $\eta=17$%. For a smaller unbalancing of $|r_{2}|/r_{1}=1.9$ (blue curve), the sensitivity is degrading faster with the added losses. Finally, if the unbalancing is too weak $|r_{2}|/r_{1}=1.5$ (green curve), as was the case already in 3b, even without adding extra losses the interferometer cannot compensate for the detector noise and the sensitivity is therefore always above the SNL.

We would like to stress that this active detection strategy involving parametric amplification can also be implemented at the output of more common SU(2) interferometers Manceau et al. (2017) (Mach-Zender, Michelson, Sagnac etc.). This is especially important in gravitational wave detectors where the detection losses considerably limit the sensitivity. Indeed, external elements of interferometers such as the Faraday isolators, output mode cleaners and photodetectors are reported as the main sources of losses Abadie J. et al (2011). Also the tolerance of the interferometer to detection losses is very important in experiments where a high phase sensitivity is required at frequency ranges (infrared, terahertz) where the efficiency of detectors is low for technical reasons. In addition, the unbalanced SU(1,1) interferometer can be reconfigured to be applied to sub-shot-noise imaging Brida et al. (2010) where the detection efficiency plays a crucial role.

In conclusion, we have demonstrated for the first time phase supersensitivity in an unseeded SU(1,1) interferometer, beating the shot noise limit by $2.3$ dB. Moreover we have shown that a gain-unbalanced interferometer is tolerant to external losses. With the parametric gain $5.2$, the SNL was beaten even for a detection efficiency as low as $17\%$. This result is relevant to many cases where detection losses are high, such as gravitational-wave detection and phase measurements in the infrared and terahertz ranges.

We acknowledge the financial support of the joint DFG-RFBR project CH1591/2-1 - 16-52-12031 NNIOa

Appendix A Detection

The detectors are Hamamatsu $S3883$ p-i-n diodes with quantum efficiency $90\%$ at $800$ nm, followed by pulsed charge-sensitive amplifiers based on Amptek A250 and A275 chips, with peaking time $2.77$ $\mu$s. For each input light pulse, the output voltage pulse has a 8 $\mu$s duration and the area depending on the input number of photons. The fundamental beam of the Spectra Physics Spitfire Ace system at $\lambda=800$ nm is used to calibrate the detector. Figure 5a shows the mean area of the output voltage pulse for a given mean number of incoming photons per pulse. For each set of measurements presented in this work, the mean number of photons is derived from this calibration curve. One can see in Fig. 5a that the response of the detector to the light pulses is linear within most of the photon-number range under study in this work. However, as it is apparent in the inset of Fig. 5a for low photon fluxes ($\lesssim 2000$ photons per pulse), a nonlinearity, with a corresponding decrease in the detection sensitivity of the detector, needs to be taken into account.

Also, the detector brings some additional noise to the measurements. The corresponding noise in terms of photons per pulse is $\Delta N_{d}=290$ photons in the linear part of the curve and considerably larger in its nonlinear part. The measurements presented in the main text are characterized by mean numbers of photons within a range of a few hundreds to a thousand in the region of interest for supersensitivity, i.e. around the dark fringe. A detector noise of $\Delta N_{d}\simeq 1000$ photons per pulse needs to be taken into account in this case.

Appendix B Period of interferences and visibility

The interference period $D$ is Pérez et al. (2014)

[TABLE]

with $\lambda_{p}$ the pump wavelength and $\delta n_{air}=n_{air}(400nm)-n_{air}(800nm)$ the dispersion of the air gap between the two crystals for the considered pump and signal wavelengths. The laboratory temperature and humidity conditions give $D=52$ mm. The phase $\phi$ can therefore be determined from $D$ and the relative distance between the two crystals. A phase shift $\phi=\pi$ is equivalent to a $52$ mm distance between the two crystals.

Figure 6 shows typical fringe patterns for two sets of parametric gain values. The distance between the crystals and the phase are arbitrarily set such that $\phi=0$ corresponds to the dark fringe. A fit with Eq. (8) gives a visibility $\mathcal{V}=97\%$ in both cases, as well as for all the measurements reported in this article. The main reason for the visibility not exceeding $97$% is that the interference is observed for spatially separated crystals. In this case, emission from both of them is partially distinguishable and a $100$% visibility can be achieved only by selecting a single plane wave, which would be the case for an infinitely small pinhole, $\Delta\theta=0$.

Appendix C Curve fitting, gain values

Following the calculations presented in Ref. Manceau et al. (2017), the interferometer output mean number of photons $\langle N_{f}\rangle$ is given by

[TABLE]

with $\nu$ being the transmission of the squeezed mode through the spatial and frequency filters with bandwidth $\Delta\theta$, $\Delta\lambda$, $\eta$ the detection efficiency of the setup in the $\Delta\theta$, $\Delta\lambda$ bandwidths, $\mu$ the interferometer internal transmission and

[TABLE]

As we work with high gain values, $\sinh r_{1}\simeq\cosh r_{1}\simeq\exp(r_{1})/2$ and $\sinh|r_{2}|\simeq\cosh|r_{2}|\simeq\exp(|r_{2}|)/2$ , which leads to a simplification of Eq. (4),

[TABLE]

Moreover, the mean number of photons inside the interferometer within the $\Delta\lambda$, $\Delta\theta$ bandwidths considered is

[TABLE]

Therefore,

[TABLE]

The number of photons in the interferometer $\langle N(\Delta\lambda,\Delta\theta)\rangle$ is measured with an avalanche photodiode and is therefore known. The internal transmission is also known, $\mu=0.97$, as well as the detection efficiency $\eta=0.77$. Thus the only unknown parameter in Eq. (8) is $|r_{2}|$. The phase independent term $\eta\nu(1-\mu)\exp(2|r_{2}|)/4$ can be incorporated into a more general term, $B$, that also takes into account additional background light due to the non-perfect interference visibility:

[TABLE]

The parametric gain of the first amplifier $r_{1}$ and the transmission $\nu$ of the squeezed mode can be subsequently found via Eq. (7) and the number of photons $\langle N_{2}(\Delta\lambda,\Delta\theta)\rangle$ produced by the second crystal in the $\Delta\theta$, $\Delta\lambda$ bandwidths, which is also measured:

[TABLE]

Figure 7 shows two examples of fit with Eq. (9) for two datasets. The average pump power was set to $90$ mW for the blue curve, while it was increased to $140$ mW for the red curve. However the number of photons inside the interferometer $\langle N(\Delta\lambda,\Delta\theta)\rangle=4.8$ is the same in both cases as the phase matching of the first crystal was modified in order to keep $\langle N(\Delta\lambda,\Delta\theta)\rangle$ constant for the two measurements. The second crystal is aligned such as to maximize the number of photons in the $\Delta\lambda$, $\Delta\theta$ windows. Hence the fits with Eq. (9) yield $|r_{2}|=3.9$ for the $90$ mW pump power excitation, while the larger pump power of $140$ mW gives a larger gain of $|r_{2}|=5.2$. Furthermore we measured $\langle N_{2}(\Delta\lambda,\Delta\theta)\rangle$, the mean number of photons emitted by the second crystal within the bandwidths $\langle\Delta\lambda$,$\Delta\theta\rangle$. Using Eq. (10), we find $r_{1}=2.5$ and $\nu=0.13$ for the $90$ mW case, while $r_{1}=2.1$ and $\nu=0.29$ for the $140$ mW case. It is to be noted that different transmission $\nu$ of the squeezed mode are found for the two measurements because the mode size changes while changing the phase matching of the first crystal.

Appendix D Pump pulses post-selection

The pump beam possesses a relatively good intensity stability with a pulse energy fluctuation around $10$ nJ for the mean pulse energy of $20$ $\mu$J. The corresponding second-order correlation function at zero delay is $g^{(2)}(0)=\frac{\langle I^{2}\rangle}{\langle I\rangle^{2}}\approx 1.00001$, with $I$ being the measured pump beam intensity. The pump intensity distribution is therefore almost Poissonian ($g^{(2)}(0)\approx 1$). However because of the high parametric gains involved in the measurements, these small-scale pump intensity fluctuations are amplified in the PDC light. We use an additional BBO crystal BBOpp to enhance these fluctuations and be in the sensitive wavelengths range of the p-i-n diode used. The broadband, multimode PDC light generated by BBOpp is then measured by a second p-i-n diode PD2 similar to PD1. A post-processing window is set to discard from the measurement set all excitation pulses outside a chosen interval. Hence, the pump fluctuations are decreased and the mean number of photons for a given interferometer position can be measured precisely.