Microwave quantum illumination using a digital receiver

TL;DR

This paper demonstrates microwave quantum illumination using entangled photons and a digital receiver, showing potential advantages over classical radar in low-noise, low-power scenarios, with experimental and simulated results.

Contribution

First experimental demonstration of microwave quantum illumination with a digital receiver, highlighting practical challenges and advantages over classical methods.

Findings

Digital phase conjugate receiver outperforms classical noise radar.

Simulated perfect idler detection yields quantum advantage.

Experimental setup illuminates objects at 1 meter with entangled microwaves.

Abstract

Quantum illumination is a powerful sensing technique that employs entangled signal-idler photon pairs to boost the detection efficiency of low-reflectivity objects in environments with bright thermal noise. The promised advantage over classical strategies is particularly evident at low signal powers, a feature which could make the protocol an ideal prototype for non-invasive biomedical scanning or low-power short-range radar. In this work we experimentally investigate the concept of quantum illumination at microwave frequencies. We generate entangled fields using a Josephson parametric converter to illuminate a room-temperature object at a distance of 1 meter in a free-space detection setup. We implement a digital phase conjugate receiver based on linear quadrature measurements that outperforms a symmetric classical noise radar in the same conditions despite the entanglement-breaking signal path. Starting from experimental data, we also simulate the case of perfect idler photon number detection, which results in a quantum advantage compared to the relative classical benchmark. Our results highlight the opportunities and challenges on the way towards a first room-temperature application of microwave quantum circuits.

Equations

$$\displaystyle\hat{a}_{S,1}^{\mathrm{det}}=\sqrt{G_{S}}\Big{(}\sqrt{\eta}\hat{a}_{S}$$

$$\hat{a}_{S,0}^{\mathrm{det}}=\sqrt{G_{S}^{\mathrm{det}}}\Big{(}\hat{a}_{n}^{\mathrm{env}}+\sqrt{1-\frac{1}{G_{S}^{\mathrm{det}}}}\hat{a}_{n,S}^{\mathrm{det}\dagger}\Big{)},$$

$${\mathrm{SNR}}_{QI/Cl}=\frac{(⟨{\hat{N}}_1⟩-⟨{\hat{N}}_0⟩{)}^2}{2{\left(\sqrt{(\Delta N_1{)}^2}+\sqrt{(\Delta N_0{)}^2}\right)}^2},$$

$$\mathrm{SNR}_{\mathrm{CS}}^{\mathrm{hom}}=\frac{(\langle\hat{X}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{X}_{S,0}^{\mathrm{det}}\rangle)^{2}}{2\Big{(}\sqrt{(\Delta X_{S,1}^{\mathrm{det}})^{2}}+\sqrt{(\Delta X_{S,0}^{\mathrm{det}})^{2}}\Big{)}^{2}},$$

$$\mathrm{SNR}_{\mathrm{CS}}^{\mathrm{het}}=\frac{(\langle\hat{X}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{X}_{S,0}^{\mathrm{det}}\rangle)^{2}+(\langle\hat{P}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{P}_{S,0}^{\mathrm{det}}\rangle)^{2}}{2\Big{(}\sum\limits_{i=1}^{2}\sqrt{(\Delta X_{S,i}^{\mathrm{det}})^{2}+(\Delta P_{S,i}^{\mathrm{det}})^{2}}\Big{)}^{2}},$$

$$N_{i}=\hbar\omega_{i}BRG_{i}\big{[}(1/2)\mathrm{coth}[\hbar\omega_{i}/(2k_{B}T)]+n_{\mathrm{nadd,i}}\big{]},$$

$$(G_S,G_I)=(93.98(01),94.25(02))\mathrm{dB}$$

$$(n_{\mathrm{add},S},n_{\mathrm{add},I})=(9.61(04),14.91(1)).$$

$${\hat{a}}_S^{\mathrm{in}}$$

$${\hat{a}}_I^{\mathrm{out}}$$

$$\hat{a}_{I}^{\mathrm{det}}=\sqrt{G_{I}}\Big{(}\hat{a}_{I}+\sqrt{\frac{G_{I}^{\mathrm{amp}}-1}{G_{I}^{\mathrm{amp}}}}\hat{a}_{n,I}^{\mathrm{amp\dagger}}+\sqrt{\frac{G_{I}^{\mathrm{det}}-1}{G_{I}}}\hat{a}_{n,I}^{\mathrm{det\dagger}}\Big{)},$$

$$n_{\mathrm{add},I}=\frac{G_{I}^{\mathrm{amp}}-1}{G_{I}^{\mathrm{amp}}}\Big{(}\langle\hat{a}_{n,I}^{\mathrm{amp\dagger}}\hat{a}_{n,I}^{\mathrm{amp}}\rangle+1\Big{)}+\frac{G_{I}^{\mathrm{det}}-1}{G_{I}^{\mathrm{amp}}G_{I}^{\mathrm{det}}}\Big{(}\langle\hat{a}_{n,I}^{\mathrm{det\dagger}}\hat{a}_{n,I}^{\mathrm{det}}\rangle+1\Big{)}=14.91(1),$$

$${\hat{a}}_{S,1}^{\mathrm{out}}$$

$${\hat{a}}_{S,0}^{\mathrm{out}}$$

$${\hat{a}}_{S,i}^{\det }=(\sqrt{G_S^{\det }}{\hat{a}}_{S,i}^{\mathrm{out}}+\sqrt{G_S^{\det }-1}{\hat{a}}_{n,S}^{\det †}),$$

$$\hat{a}_{S,1}^{\mathrm{det}}=\sqrt{G_{S}}\Big{(}\sqrt{\eta}\hat{a}_{S}+\sqrt{\frac{\eta(G_{S}^{\mathrm{amp}}-1)}{G_{S}^{\mathrm{amp}}}}\hat{a}_{n,S}^{\mathrm{amp\dagger}}+\sqrt{\frac{1-\eta}{G_{S}^{\mathrm{amp}}}}\hat{a}_{n}^{\mathrm{env}}+\sqrt{\frac{G_{S}^{\mathrm{det}}-1}{G_{S}}}\hat{a}_{n,S}^{\mathrm{det\dagger}}\Big{)},$$

$$\hat{a}_{S,0}^{\mathrm{det}}=\sqrt{G_{S}^{\mathrm{det}}}\Big{(}\hat{a}_{n}^{\mathrm{env}}+\sqrt{1-\frac{1}{G_{S}^{\mathrm{det}}}}\hat{a}_{n,S}^{\mathrm{det}\dagger}\Big{)},$$

$$n_{\mathrm{add},S}=\frac{G_{S}^{\mathrm{amp}}-1}{G_{S}^{\mathrm{amp}}}\Big{(}\langle\hat{a}_{n,S}^{\mathrm{amp\dagger}}\hat{a}_{n,S}^{\mathrm{amp}}\rangle+1\Big{)}+\frac{G_{S}^{\mathrm{det}}-1}{G_{S}^{\mathrm{amp}}G_{S}^{\mathrm{det}}}\Big{(}\langle\hat{a}_{n,S}^{\mathrm{det\dagger}}\hat{a}_{n,S}^{\mathrm{det}}\rangle+1\Big{)}=9.61(04),$$

$${\hat{a}}_{i,\pm }\equiv \frac{{\hat{a}}_{S,i}^{PC}\pm {\hat{a}}_I^{\det }}{\sqrt{2}}.$$

$${\hat{N}}_i={\hat{N}}_{i,+}-{\hat{N}}_{i,-}.$$

$${\mathrm{SNR}}_{QI/Cl}=\frac{(⟨{\hat{N}}_1⟩-⟨{\hat{N}}_0⟩{)}^2}{2{\left(\sqrt{(\Delta N_1{)}^2}+\sqrt{(\Delta N_0{)}^2}\right)}^2},$$

$${\mathrm{SNR}}_{QI/Cl}=\frac{[(⟨{\hat{N}}_{1,+}⟩-⟨{\hat{N}}_{1,-}⟩)-(⟨{\hat{N}}_{0,+}⟩-⟨{\hat{N}}_{0,-}⟩){]}^2}{2{\left(\sqrt{(\Delta N_1{)}^2}+\sqrt{(\Delta N_0{)}^2}\right)}^2},$$

$$⟨{\hat{N}}_{0,+}⟩-⟨{\hat{N}}_{0,-}⟩$$

$$⟨{\hat{N}}_{1,+}⟩-⟨{\hat{N}}_{1,-}⟩$$

$$(\Delta N_i{)}^2=⟨{\hat{N}}_{i,+}⟩(⟨{\hat{N}}_{i,+}⟩+1)+⟨{\hat{N}}_{i,-}⟩(⟨{\hat{N}}_{i,-}⟩+1)-(⟨{\hat{a}}_{S,i}^{PC†}{\hat{a}}_{S,i}^{PC}⟩-⟨{\hat{a}}_I^{†}{\hat{a}}_I⟩{)}^2/2,$$

$$\mathrm{SNR}_{\mathrm{CS}}^{\mathrm{het}}=\frac{(\langle\hat{X}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{X}_{S,0}^{\mathrm{det}}\rangle)^{2}+(\langle\hat{P}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{P}_{S,0}^{\mathrm{det}}\rangle)^{2}}{2\Big{(}\sqrt{(\Delta X_{S,1}^{\mathrm{det}})^{2}+(\Delta P_{S,1}^{\mathrm{det}})^{2}}+\sqrt{(\Delta X_{S,0}^{\mathrm{det}})^{2}+(\Delta P_{S,0}^{\mathrm{det}})^{2}}\Big{)}^{2}}.$$

$$\mathrm{SNR}_{\mathrm{CS}}^{\mathrm{hom}}=\frac{(\langle\hat{X}_{S,1}^{\mathrm{det}}\rangle-\langle\hat{X}_{S,0}^{\mathrm{det}}\rangle)^{2}}{2\Big{(}\sqrt{(\Delta X_{S,1}^{\mathrm{det}})^{2}}+\sqrt{(\Delta X_{S,0}^{\mathrm{det}})^{2}}\Big{)}^{2}}.$$

Full text

Quantum sensing is well developed for photonic applications SensingReview inline with other advanced areas of quantum information NielsenChuang ; Weedbrook12 ; Hayashi17 ; Watrous18 . As a matter of fact, quantum optics has been so far the most natural and convenient setting for implementing the majority of protocols in quantum communication, cryptography and metrology Braunstein94 . The situation is different at longer wavelengths, such as THz or microwaves, for which the current variety of quantum technologies is more limited and confined to cryogenic environments. With the exception of superconducting quantum processing Schoelkopf08 no microwave quanta are typically used for applications such as sensing and communication. For such tasks high energy and low loss optical and telecom frequency signals represent the first choice and form the communication backbone in the future vision of a hybrid quantum internet Kimble2008 ; PirBra16 ; Wehner18 .

Despite this general picture, there are applications of quantum sensing that are naturally embedded in the microwave regime. This is exactly the case with quantum illumination (QI) Lloyd08 ; Tan08 ; Lopaeva2013 ; Zhang2013 ; Zhang2015 ; Weedbrook2016 ; Shapiro2019 for its remarkable robustness to background noise, which at room temperature amounts to $\sim 10^{3}$ thermal quanta per mode at a few GHz. In QI, the aim is to detect a low-reflectivity object in the presence of very bright thermal noise. This is accomplished by probing the target with less than one entangled photon per mode, in a stealthy non-invasive fashion which is impossible to reproduce with classical means. In the Gaussian QI protocol Tan08 , the light is prepared in a two mode squeezed vacuum state Weedbrook12 with the signal mode sent to probe the target while the idler mode is kept at the receiver. Even though entanglement is lost in the round trip from the target, the surviving signal-idler correlations, when appropriately measured, can be strong enough to beat the performance achievable by the most powerful classical detection strategy. In the low photon flux regime, where QI shows the biggest advantage, it could be suitable for extending quantum sensing techniques to short-range radar Choi2016 and non-invasive diagnostic scanner applications Lin1992 .

Previous experiments in the microwave domain Wilson19 ; Wilson1903 demonstrated a quantum enhancement of the detected covariances compared to a symmetric classical noise radar, i.e. with approximately equal signal and idler photon number. With appropriate phase sensitive detection an ideal classically correlated noise radar can be on par or, in the case of a bright idler Shapiro2019 , even outperform coherent heterodyne detection schemes, which maximize the signal-to-noise (SNR) ratio for realistic (phase-rotating) targets. However, if the phase of the reflected signal is stable over relevant timescales or a priori known, homodyne detection represents the strongest classical benchmark.

In this work, we implement a digital version of the phase-conjugated receiver of Ref. Guha , experimentally investigating proof of concept QI in the microwave regime Barzanjeh15 . We use a Josephson parametric converter (JPC) Bergeal2010 ; Abdo2013 inside a dilution refrigerator for entanglement generation Flurin2012 ; Silveri2015 . The generated signal microwave mode, with annihilation operator $\hat{a}_{S}$, is amplified to facilitate its detection and sent to probe a room-temperature target, while the idler mode $\hat{a}_{I}$ is measured as schematically shown in Fig. 1(a). The reflection from the target $\hat{a}_{R}$ is also detected and the two measurement results are post-processed to calculate the SNR for discriminating the presence or absence of the object. Our experimental implementation of QI relies on linear quadrature measurements and suitable post-processing in order to compute all covariance matrix elements from the full measurement record as shown in previous microwave quantum optics experiments with linear detectors daSilva10 ; Bozyigit11 ; Eichler12 . This enables an implementation of the phase-conjugated receiver that fully exploits the correlations of the JPC output fields without analog photodetection. We then compare the SNR with other detection strategies for the same signal path, i.e. the same signal photon numbers at the JPC output, which is also our reference point for the theoretical modeling.

Our digital approach to QI circumvents common practical problems such as finite idler storage time that can limit the range and fidelity of QI detection schemes. However, this advantage comes at the expense that the theoretically strongest classical benchmark in the same conditions - the coherent state homodyne detector using the same signal power and signal path - can be approached in specific conditions such as quantum limited amplification, but never be outperformed. To outperform coherent state homodyne detection in practice, will require low temperature square law detection of microwave fields that can be realized with radiometer or photon counting measurements. Nevertheless, using calibration measurements of the idler path, we can simulate a situation with perfect idler photon number detection, extrapolating the case where the reflected mode is detected together with the idler mode using analog microwave photon counters. For this situation we show that the SNR of coherent heterodyne detection and symmetric noise radars is exceeded by up to 4 dB and that of homodyne detection - the classical benchmark - by up to 1 dB for the same amplified signal path, measurement bandwidth and signal power. We also note that the strong and noisy amplification of the signal path chosen to facilitate the detection with commercial analog-to-digital converters enables another classical receiver strategy, i.e. the detection of the amplifier noise in the presence of the target. Since the amplified noise exceeds the environmental noise at room temperature by orders of magnitude, this would indeed be the most effective strategy for the implemented experiment. For the same reason, a low noise coherent source at room temperature would outperform the relative benchmarks considered here. In practice, outperforming the room temperature benchmark depends on the chosen amount of gain, the type of amplifier and the loss in the detection system and therefore does not pose a fundamental limitation to the presented measurement scheme that focusses on the relative comparison of the different illumination types.

The experimental setup, shown in Fig. 1(b), is based on a frequency tunable superconducting JPC operated in the three-wave mixing regime and pumped at the sum of signal and idler frequencies $\omega_{p}=\omega_{S}+\omega_{I}$, see Methods for more details. The output of the JPC contains a nonzero phase-sensitive cross correlation $\langle\hat{a}_{S}\hat{a}_{I}\rangle$, which leads to entanglement between the signal mode with frequency $\omega_{S}=10.09$ GHz and the idler mode with frequency $\omega_{I}=6.8$ GHz. In our work, the quantities $\langle\hat{O}\rangle$ and $(\Delta O_{i})^{2}=\langle\hat{O}_{i}^{2}\rangle-\langle\hat{O}_{i}\rangle^{2}$ define the mean and the variance of the operator $\hat{O}$, respectively, and they are evaluated from experimental data. The signal and idler are sent through two different measurement lines, where they are amplified, filtered, down-converted to an intermediate frequency of $20$ MHz and digitized with a sampling rate of 100 MHz using an 8 bit analog-to-digital converter. Applying fast Fourier transform and post-processing to the measured data, we obtain the quadrature voltages $I_{i}$ and $Q_{i}$, which are related to the complex amplitudes $a_{i}$ and their conjugate $a_{i}^{*}$ of the signal and idler modes at the outputs of the JPC as $a_{i}=\frac{I_{i}+\mathrm{i}\,Q_{i}}{\sqrt{2\hbar\omega_{i}BRG_{i}}}$ and $a_{i}^{*}=\frac{I_{i}-\mathrm{i}\,Q_{i}}{\sqrt{2\hbar\omega_{i}BRG_{i}}}$, having the same measurement statistics as the annihilation operator $\hat{a}_{i}$. Here, $R=50\,\Omega$, $B=200$ kHz is the measurement bandwidth set by a digital filter and $i=S,I$ Menzel ; Eichler12 ; Barzanjeh19 . We calibrate the system gain $(G_{S},G_{I})=(93.98(01),94.25(02))$ dB and system noise $(n_{\mathrm{add},S},n_{\mathrm{add},I})=(9.61(04),14.91(1))$ of both measurement channels as described in Methods.

A first important check for the experiment is to quantify the amount of entanglement at the output of the JPC at 7 mK. A sufficient condition for the signal and idler modes to be entangled is the non-separability criterion Duan2000 $\Delta:=\langle\hat{X}^{2}_{-}\rangle+\langle\hat{P}^{2}_{+}\rangle<1$, for the joint field quadratures $\hat{X}_{-}=(\hat{a}_{S}+\hat{a}_{S}^{\dagger}-\hat{a}_{I}-\hat{a}_{I}^{\dagger})/2$ and $\hat{P}_{+}=(\hat{a}_{S}-\hat{a}_{S}^{\dagger}+\hat{a}_{I}-\hat{a}_{I}^{\dagger})/(2\mathrm{i})$. In Fig. 2(a) we show measurements of $\Delta$ as a function of the signal photon number $N_{S}=\langle\hat{a}_{S}^{\dagger}\hat{a}_{S}\rangle$ at the output of the JPC at millikelvin temperatures, as obtained by applying the above calibration procedure to both signal and idler modes, and compare the result with classically-correlated radiation. The latter is generated at room temperature using the white noise mode of an arbitrary waveform generator, divided into two different lines, individually up-converted to the signal $\omega_{S}$ and idler $\omega_{I}$ frequencies and fed to the JPC inside the dilution refrigerator. Note that, for both JPC and classically correlated noise, we digitally rotate the relative phase of the quadratures to maximize the correlation between signal and idler.

The classically-correlated signal and idler modes are then reflected back from the JPC (pumps are off) and pass through the measurement lines attached to the outputs of the JPC. This ensures that both classical and quantum radiations experience the same conditions in terms of gain, loss, and noise before reaching the target and before being detected in the identical way. As shown in Fig. 2(a), at low photon number the parameter $\Delta$ is below one proving that the outputs of the JPC are entangled, while at larger photon number (larger pump power) the entanglement gradually degrades and vanishes at $N_{S}=4.5$ photons $\cdot\mathrm{s^{-1}\cdot Hz^{-1}}$. We attribute this to finite losses in the JPC, which leads to pump power dependent heating and results in larger variances of the output field. The classically correlated radiation of the same signal power on the other hand (red data points), cannot fulfill the non-separability criterion and therefore $\Delta\geq 1$ for the entire range of the signal photons. In the latter case we also observe a slow relative degradation of the classical correlations as a function of the signal photon numbers, which could be improved with more sophisticated noise generation schemes Wilson19 .

The experiments of QI and classically-correlated illumination (CI) are implemented in a similar way, see Fig. 1(b). The two amplified quadratures of the idler mode $\hat{a}_{I}^{\mathrm{det}}$ are measured at room temperature, and the signal mode $\hat{a}_{S}$ is amplified (with gain $G_{S}^{\mathrm{amp}}$ and the noise mode $\hat{a}_{n,S}^{\mathrm{amp}}$) and used to probe a noisy region that is suspected to contain an object. In this process, we define $\eta$ as the total detection loss on the signal path between the two room-temperature switches used in the measurement chain, which includes cable loss, free-space loss, and object reflectivity. The reflected signal from the region is measured, by means of a mixer and an amplifier with gain $G_{S}^{\mathrm{det}}$ and the noise mode $\hat{a}_{n,S}^{\mathrm{det}}$. The output $\hat{a}_{S,i}^{\mathrm{det}}$ in the presence ($i=1$) or absence ($i=0$) of the object is then post-processed for the reconstruction of the covariance matrix of the detected signal-idler state.

The signal mode $\hat{a}_{S,i}^{\mathrm{det}}$ takes different forms depending on the presence

[TABLE]

or absence

[TABLE]

of the target with $\hat{a}_{n}^{\mathrm{env}}$ is the environmental noise mode. In the absence of the object, the signal contains only noise $n_{0}=G_{S}^{\mathrm{det}}n_{\mathrm{env}}+(G_{S}^{\mathrm{det}}-1)n_{\mathrm{det},S}$ in which $n_{\mathrm{det},S}$ is the amplifier added noise after interrogating the object region. In the presence of the target and for $\eta\ll 1$, the added noise to the signal is $n_{1}=\eta\,G_{S}^{\mathrm{det}}(G_{S}^{\mathrm{amp}}-1)n_{\mathrm{amp},S}+n_{0}$, whose first term is due to the amplifier added noise of the first amplification stage before reaching the target, which exceeds the environmental noise $n_{\mathrm{env}}$ as well as the signal photon numbers used to probe the target. This implies that in our proof of principle demonstration the optimal classical strategy would actually be based on detecting the presence or absence of the amplifier noise rather than the coherences and correlations of the signal-idler path with the measured $\mathrm{SNR}_{\mathrm{passive}}=(n_{1}-n_{0})/(n_{0}+1)\simeq 31.4\,$dB for the chosen gain and receiver noise in our setup. However, for lower noise temperature signal amplifiers and lower gain as well as in longer range applications with increased loss, such a passive signature of the detection scheme will be drastically reduced and eventually disappear in the environmental noise at room temperature.

The final step of the measurement is the application of a digital version of the phase-conjugate receiver Guha . The reflected mode $\hat{a}_{S,i}^{\mathrm{det}}$ is first phase-conjugated, and then combined with the idler mode on a 50-50 beam splitter. As described in Methods, the SNR of the balanced difference photodetection measurement reads

[TABLE]

where $\hat{N}_{i}=\hat{a}_{i,+}^{\dagger}\hat{a}_{i,+}-\hat{a}_{i,-}^{\dagger}\hat{a}_{i,-}$ with $\hat{a}_{i,\pm}=(\hat{a}_{S,i}^{\mathrm{det}\dagger}+\sqrt{2}\hat{a}_{v}\pm\hat{a}_{I}^{\mathrm{det}})/\sqrt{2}$ is the annihilation operator of the mixed signal and idler modes at the output of the beam splitter in the absence ($i=0$) and the presence ($i=1$) of the target (here $\hat{a}_{v}$ is the vacuum noise operator). For the raw SNR without idler calibration we use Eq. (3). In order to simulate perfect photon number detection of the idler mode directly at the JPC output we reduce the variance in the denominator of Eq. (3) by the calibrated idler vacuum and amplifier noise as $\langle\hat{a}_{I}^{\dagger}\hat{a}_{I}\rangle=\langle\hat{a}_{i}^{\mathrm{det}\dagger}\hat{a}_{i}^{\mathrm{det}}\rangle/G_{I}-(n_{\mathrm{add},I}+1)$, see Methods.

The experiment of coherent state illumination is performed by generating a weak coherent tone using a microwave source at room temperature followed by low temperature chain of thermalized attenuators inside the dilution refrigerator. The center frequency of the coherent tone is $\omega_{S}$, exactly matched with the frequency of the signal used in the QI and CI experiments. The coherent tone is reflected back from the unpumped JPC and directed into the same measurement chain identical to that of QI and CI, see Fig. 1(b). The signal is sent to probe a target region and the detected radiation $\hat{a}_{S,i}^{\mathrm{det}}$ is used to calculate the SNR of the digital homodyne and heterodyne detections for the same probe power, bandwidth and amplifier noise.

In the absence of a passive signature due to signal noise amplification, digital homodyne detection of a coherent state represents the optimal classical strategy in terms of the SNR, which is given by

[TABLE]

while the SNR of the digital heterodyne detection is lower and given by

[TABLE]

where $\hat{X}_{S,i}^{\mathrm{det}}=\frac{\hat{a}_{S,i}^{\mathrm{det}}+\hat{a}_{S,i}^{\mathrm{det}\dagger}}{\sqrt{2}}$ and $\hat{P}_{S,i}^{\mathrm{det}}=\frac{\hat{a}_{S,i}^{\mathrm{det}}-\hat{a}_{S,i}^{\mathrm{det}\dagger}}{i\sqrt{2}}$ are the field quadrature operators (see Methods for more details).

In Fig. 2(b) we compare the SNR of QI and CI with and without idler calibration for a perfectly reflective object in a zero loss channel $\eta=1$. For comparison, we also include the results of coherent-state illumination with homodyne and heterodyne detection. In all cases the signal mode at room temperature is overwhelmed with amplifier noise. We use 3 sets of measurements to calculate the standard deviation of the mean SNR of a single mode measurement with measurement time $T=1/B=5\,\mu$s. Each set is based on $M=380\,$k samples ($192\,$k for the coherent state detection) corresponding to a measurement time of 1.87 seconds (0.93 seconds for the coherent state detection). To get the total statistics the measurement time takes 5.6 seconds (2.8 seconds for the coherent state detection). For the same measurement bandwidth and using the raw data of the measured quadrature pairs (solid lines) QI (blue dots) outperforms sub-optimum symmetric CI (red dots) by up to 3 dB at low signal photon numbers but it cannot compete with the SNR obtained with coherent state illumination (yellow and green dots). Under the assumption of perfect idler photon number detection, i.e. applying the calibration discussed above (dashed lines), the SNR of QI is up to 4 dB larger than that of symmetric CI and coherent-state illumination with heterodyne detection, which does not require phase information, over the region where the outputs of the JPC are entangled. For signal photon numbers $N_{S}>4.5$, where there is no entanglement present in the signal source, the sensitivity of the coherent-state transmitter with heterodyne detection outperforms QI and CI, confirming the critical role of entanglement to improve the sensitivity of the detection.

QI with a phase conjugate receiver is potentially able to outperform coherent-state illumination with homodyne detection by up to 3 dB, i.e. the optimum classical benchmark, in the regime of low signal photon numbers. In the region $N_{S}<0.4$ the experimentally inferred SNR of QI is approximately 1 dB larger, in agreement with the theoretical prediction taking into account experimental non-idealities like the finite squeezing of the source. In practice though, i.e. without the applied idler calibration, the quantum advantage compared to coherent homodyne detection is not accessible with a digital receiver based on heterodyne measurements, even in the case of quantum limited amplifiers, due to the captured idler vacuum noise, which lowers the optimal SNR by at least 3 dB Tan08 ; Weedbrook2016 . The experimental results (dots) are in very good agreement with the theoretical prediction (solid and dashed lines). For the theory we rewrite the SNRs Eqs. (3)-(5) in terms of the signal photon number $N_{S}=\langle\hat{a}_{S}^{\dagger}\hat{a}_{S}\rangle$, the idler photon number $N_{I}=\langle\hat{a}_{I}^{\dagger}\hat{a}_{I}\rangle$, and the signal-idler correlation $\langle\hat{a}_{S}\hat{a}_{I}\rangle$ at the output of the JPC. These parameters are extracted from the measured and calibrated data as a function of the JPC pump power. Together with the known system gain and noise we plot the theoretical predictions of the various protocols at room-temperature.

An important feature of a radar or short range scanner is its resilience with respect to signal loss. To verify this, as shown in Fig. 1(b), we use two microwave switches at room temperature in the signal line in order to select between a digitally controllable step attenuator to mimic an object with tunable reflectivity and a proof of principle radar setup. With this setup we determine the effects of loss and object reflectivity as well as target distance on the efficiency of the quantum enhanced radar. In Fig. 3(a) we plot the measured SNR of QI, CI and coherent-state illumination with heterodyne detection, as a function of the imposed loss on the signal mode. The calibrated QI protocol is always superior to calibrated symmetric CI and coherent-state illumination with heterodyne detection for a range of effective loss -25 dB $<\eta<0$ dB. The dashed lines are the theory predictions from Eq. (3) and Eq. (5) for a fixed chosen signal photon number $N_{S}=0.5$. The shaded regions represent the confidence interval extracted from the standard deviation of the measured idler photon numbers and the cross-correlations as a function of $\eta$.

In the context of radar, small improvements in the SNR lead to the exponentially improved error probability $\mathcal{E}=1/2\,\mathrm{erfc}{(\sqrt{\textrm{SNR}\cdot M})}$, where $M=T_{\mathrm{tot}}B$ is the number of single mode measurements, and $T_{\mathrm{tot}}$ is the total measurement time required for a successful target detection. To test the principle of microwave QI in free-space at room temperature, we amplify and send the microwave signal emitted from JPC to a horn antenna and a copper plate representing the target at a variable distance. The reflected signal from this object is collected using a second antenna of the same type, down-converted, digitized, and combined with the calibrated idler mode to calculate the SNR of the binary decision. With this setup we repeat the measurement for CI and coherent state illumination with heterodyne. Fig. 3(b) shows the SNR of these protocols as a function of the object distance from the transmitting antenna as well as the total loss of the free space link. Calibrated QI reveals higher sensitivity for a reflective target up to 1 meter away from the transmitting antenna. The results are in good agreement with the theoretical model.

Conclusion

In this work we have studied proof of concept quantum illumination in the microwave domain, the most natural frequency range for target detection. Assuming perfect idler photon number detection we showed that a quantum advantage is possible despite the entanglement-breaking signal path. Since the best results are achieved for less than one mean photon per mode, our experiment indicates the potential of QI as non-invasive scanning method, e.g. for biomedical applications, imaging of human tissues or non-destructive rotational spectroscopy of proteins, besides its potential use as short-range low-power radar, e.g. for security applications. However, for this initial proof of principle demonstration the amplified bright noise in the target region overwhelms the environmental noise by orders of magnitude, which precludes the non-invasive character at short target distances and presents an opportunity to use the presence or absence of the amplifier noise to detect the object with even higher SNR. The use of quantum limited parametric amplifiers Yurke1989 ; Beltran2007 ; Macklin2015 with limited gain, such that the amplified vacuum does not significantly exceed environmental or typical electronic noise at the target, will help to achieve a practical advantage with respect to the lowest noise-figure coherent state heterodyne receivers at room temperature and, up to the vacuum noise, they will also render the idler calibration obsolete. The use of sensitive radiometers or microwave single photon detectors Kono ; Besse ; Lescanne at millikelvin temperatures without signal amplification, represents a promising route to achieve an advantage in practical situations and with respect to ideal coherent state homodyne receivers. One advantage of the presented digital implementation of QI is that it does not suffer from the idler storage problem of receivers that rely on analog photodetection schemes, inherently limiting the accessible range when used as a radar. It is an interesting open question what other types of receivers Zhuang2017 could be implemented in the microwave domain, based on state of the art superconducting circuit technology and digital signal processing.

Acknowledgments We thank Chris Wilson and in particular Jeffrey Shapiro for helpful critical comments and discussions. We thank IBM for donating the JPC used in this work, Jerry Chow and Baleegh Abdo for helpful advice, as well as Mariia Labendik and Riya Sett for their contribution to characterizing the JPC properties.

Funding: This work was supported by the Institute of Science and Technology Austria (IST Austria), the European Research Council under grant agreement number 758053 (ERC StG QUNNECT) and the EU’s Horizon 2020 research and innovation programme under grant agreement number 862644 (FET Open QUARTET). S.B. acknowledges support from the Marie Skłodowska Curie fellowship number 707438 (MSC-IF SUPEREOM), DV acknowledge support from EU’s Horizon 2020 research and innovation programme under grant agreement number 732894 (FET Proactive HOT) and the Project QuaSeRT funded by the QuantERA ERANET Cofund in Quantum Technologies, and J.M.F from the Austrian Science Fund (FWF) through BeyondC (F71), a NOMIS foundation research grant, and the EU’s Horizon 2020 research and innovation programme under grant agreement number 732894 (FET Proactive HOT).

Author Contributions: S.B. S.P., and D.V. proposed and developed the theoretical idea. S.B. and J.M.F conceived the experiment, built the experimental setup and analyzed the data. S.B. performed the theoretical calculation and the measurements. All authors contributed to the manuscript.

Competing interests: The authors declare no competing interests.

Supplementary Information

I Josephson parametric converter

We use a nondegenerate three-wave mixing Josephson parametric converter (JPC) that acts as a nonlinear quantum-limited amplifier whose signal, idler and pump ports are spatially separated, as shown in Fig. 4. The nonlinearity of the JPC originates from a Josephson ring modulator (JRM) consisting of four Josephson junctions arranged on a rectangular ring and four large shunting Josephson junctions inside the ring Abdo17 . The total geometry supports two differential and one common mode. The correct bias point is selected by inducing a flux in the JRM loop by using an external magnetic field. The two pairs of the microwave half-wavelength microstrip transmission line resonators connected to the center of JRM serve as signal and idler microwave resonators. These resonators are coupled to two differential modes of the JRM and capacitively attached to two external feedlines, coupling in and out the microwave signal to the JPC.

The entanglement between signal mode with frequency $\omega_{S}$ and idler mode with frequency $\omega_{I}$ is generated by driving the non-resonant common mode of the JRM at frequency $\omega_{p}=\omega_{I}+\omega_{S}$. Two off-chip, broadband 180 degree hybrids are used to add the idler or signal modes to the pump drive. In our configuration we apply the pump to the idler side and terminate the other port of the signal hybrid with a $50\,\Omega$ cold termination. The frequency of the signal mode is $\omega_{S}=10.09$ GHz and the frequency of the idler mode is $\omega_{I}=6.8$ GHz. The maximum dynamical bandwidth and gain of our JPC are $20$ MHz and $30$ dB, respectively. The 1 dB compression point corresponds to the power $-128$ dBm at the input of the device at which the device gain drops by 1 dB and the amplifier starts to saturate. The frequency of the signal and idler modes can be varied over 100 MHz span by applying a dc current to the flux line.

II Noise calibration

The system gain $G_{i}$ and system noise $n_{\mathrm{add},i}$ of both signal and idler measurement chains are calibrated by injecting a known amount of thermal noise using two temperature controlled $50\,\Omega$ cold loads Flurin2012 ; Ku2015 . The calibrators are attached to the measurement setup with two copper coaxial cables of the same length and material as the cables used to connect the JPC via two latching microwave switches (Radiall R573423600). A thin copper braid was used for weak thermal anchoring of the calibrators to the mixing chamber plate. By measuring the noise density in V2/Hz at each temperature as shown in Fig. 5, and fitting the obtained data with the expected scaling

[TABLE]

where $B=200$ kHz and $R=50\,\Omega$, we accurately back out the total gain

[TABLE]

and the number of added noise photons referenced to the JPC output

[TABLE]

The 95% confidence values are taken from the standard error of the fit shown in Fig. 5.

III Measurement chain: gain and added noise

In Fig. 6 we show the full measurement chain used in our experiment. The outputs of the JPC, the signal $\hat{a}_{S}$ and the idler $\hat{a}_{I}$, pass through two separate superconducting lines and are amplified individually using two high electron mobility transistor (HEMT model LNF) amplifiers at the 4 K temperature stage and amplified once more at room temperature. The total gain of the amplifier chain is $G_{i}^{\mathrm{amp}}$. The output of the amplifiers for $G_{i}^{\mathrm{amp}}\gg 1$ are

[TABLE]

where $\hat{a}_{n,i}^{\mathrm{amp}}$ with $i=S,I$ is the annihilation operator of the noise mode added by the HEMT and one additional room temperature amplifier (not shown) and the preceding cable and connector losses. The idler mode is then down-converted to 20 MHz, filtered, amplified using an amplifier with gain $G_{I}^{\mathrm{det}}$ and noise annihilation operator $\hat{a}_{n,I}^{\mathrm{det}}$, and recorded using an 8 bit analog to digital card (ADC). The down-converted and detected idler mode is related to the idler mode right after the JPC as

[TABLE]

where $G_{I}=G_{I}^{\mathrm{det}}G_{I}^{\mathrm{amp}}=94.25(02)$ dB is the total gain and

[TABLE]

are the total added noise quanta referenced to the JPC output.

The signal mode is used to probe the target region. The reflected signal from the target region in the presence $H_{1}$ or absence $H_{0}$ of the target, respectively, is given by

[TABLE]

where $\eta$ is the total signal loss and $\hat{a}_{n}^{\mathrm{env}}$ is the annihilation operator of the environmental noise mode at room temperature. In the case of free space illumination we realize the absence of the target by removing the target in front of the antennas, while in the case of using a step-attenuator we mimic the absence of the target by using a 50 $\Omega$ load at the RF port of the mixer.

The signal mode after down conversion is given by

[TABLE]

with $i=0,1$, $G_{S}^{\mathrm{det}}$ is the gain and $\hat{a}_{n,S}^{\mathrm{det}}$ is the noise operator of the amplification stage after down conversion. Substituting Eqs. (III), (12a) and (12b) into Eq. (13) gives the detected signal mode in the target-presence

[TABLE]

or target-absence

[TABLE]

where $G_{S}=G_{S}^{\mathrm{det}}G_{S}^{\mathrm{amp}}=93.98(01)$ dB is the total gain with $G_{S}^{\mathrm{det}}=16.82$ dB, $G_{S}^{\mathrm{amp}}=77.16$ dB, and

[TABLE]

are the total added noise quanta at the JPC output. The total added noise in the presence of the target is given by $n_{1}=\eta\,G_{S}^{\mathrm{det}}(G_{S}^{\mathrm{amp}}-1)n_{\mathrm{amp},S}+(1-\eta)G_{S}^{\mathrm{det}}n_{\mathrm{env}}+(G_{S}^{\mathrm{det}}-1)n_{\mathrm{det},S}$ which, in the limit of $\eta\ll 1$, leads to $n_{1}=\eta\,G_{S}^{\mathrm{det}}(G_{S}^{\mathrm{amp}}-1)n_{\mathrm{amp},S}+n_{0}$, where $(G_{S}^{\mathrm{amp}}-1)n_{\mathrm{amp},S}\approx 5\times 10^{8}$. The total added noise in the absence of the target is $n_{0}=G_{S}^{\mathrm{det}}n_{\mathrm{env}}+(G_{S}^{\mathrm{det}}-1)n_{\mathrm{det},S}$, where $n_{\mathrm{env}}=\langle\hat{a}_{n}^{\mathrm{env}\dagger}\hat{a}_{n}^{\mathrm{env}}\rangle=672$ is the environmental thermal noise of the room-temperature object and $n_{\mathrm{det},S}=\langle\hat{a}_{\mathrm{n},S}^{\mathrm{det}\dagger}\hat{a}_{\mathrm{n},S}^{\mathrm{det}}\rangle+1\approx 3\times 10^{5}$ is the receiver noise dominated by the amplifier noise after down-conversion to the intermediate frequency.

IV Digital post-processing

In this section we explain how the single-mode post-processing was performed. As shown in Fig. 7(a), the down-converted and amplified signal and idler modes are continuously recorded with 100 MS/s using a two-channel ADC with 8 bit resolution. The total measurement time of the QI/CI detections (coherent-state detections) is $5.76$ seconds ($2.88$ seconds) in which the recorded data are chopped to $M=1.15\times 10^{6}\,(6\times 10^{5})$ records, each contains $500$ samples which corresponds to a filter bandwidth of 200 kHz. The $500$ samples are used to perform fast Fourier transform (FFT) on each record individually and extract the complex quadrature voltages $I_{I}$, $Q_{I}$ and $I_{S}$, $Q_{S}$ of the intermediate frequency component at 20 MHz. We calculate the detected field quadratures of both signal and idler modes $X_{i}^{\mathrm{det}}=I_{i}/\sqrt{\hbar\omega_{i}BR}$ and $P^{\mathrm{det}}_{i}=Q_{i}/\sqrt{\hbar\omega_{i}BR}$ with $i=S,I$ for $M$ measurement results, which have the same measurement statistics as the quadrature operators $\hat{X}_{i}^{\mathrm{det}}$ and $\hat{P}_{i}^{\mathrm{det}}$, where $\hat{a}^{\mathrm{det}}_{i}=(\hat{X}^{\mathrm{det}}_{i}+\mathrm{i}\,\hat{P}^{\mathrm{det}}_{i})/\sqrt{2}$.

V Digital phase-conjugate receiver: QI and CI

Both the JPC and a correlated classical source generate a zero-mean, two-mode Gaussian state with a nonzero cross correlation $\langle\hat{a}_{S}\hat{a}_{I}\rangle=\langle\hat{a}_{S}^{\mathrm{det}}\hat{a}_{I}^{\mathrm{det}}\rangle/\sqrt{G_{S}G_{I}}$. To quantify this correlation, $M$ copies of the measurement results with the statistics of $\hat{a}_{S}^{\mathrm{det}}$ and $\hat{a}_{I}^{\mathrm{det}}$ are sent individually through the digital phase-conjugate receiver in which we first perform phase-conjugation on the received individual signal $\hat{a}_{S,i}^{PC}=\sqrt{2}\hat{a}_{v}+\hat{a}_{S,i}^{\mathrm{det}\dagger}$ ($\hat{a}_{v}$ is the vacuum operator) and then mix it with the retained corresponding idler modes on a 50–50 beam splitter, as shown in Fig. 7(b), whose outputs are

[TABLE]

The target absence-or-presence decision is made by comparing the difference of the two detectors’ total photon counts Barzanjeh15 , which is equivalent to the measurement of the operator

[TABLE]

where $\hat{N}_{i,\pm}\equiv\hat{a}_{i,\pm}^{\dagger}\hat{a}_{i,\pm}$. Since our QI protocol employs a large number of copies $M$, the central limit theorem implies that the measurement of $\sum_{k=1}^{M}\hat{N}_{i,\pm}^{(k)}$ yields a random variable that is Gaussian, conditioned on target absence or target presence. It follows that the receiver’s SNR for QI or CI satisfies

[TABLE]

with $\langle\hat{O}_{i}\rangle$ and $(\Delta O_{i})^{2}=\langle\hat{O}_{i}^{2}\rangle-\langle\hat{O}_{i}\rangle^{2}$, for $i=0,1$, being the conditional means and conditional variances of $\hat{O}_{i}$, respectively, and the brackets $\langle...\rangle$ denote an average over all of the $M$ copies. For the reported raw SNR we use Eq. (19) without any calibration applied. To infer the hypothetical SNR that could be obtained with access to the idler mode directly at the JPC output $\hat{a}_{I}$, we rewrite Eq. (19) in terms of single-mode moments, i.e.,

[TABLE]

where

[TABLE]

and Guha

[TABLE]

for $i=0,1$, where we take the calibrated noiseless idler photon number $\langle\hat{a}_{I}^{\dagger}\hat{a}_{I}\rangle=\langle\hat{a}_{I}^{\mathrm{det}\dagger}\hat{a}_{I}^{\mathrm{det}}\rangle/G_{I}-(n_{\mathrm{add},I}+1)$. Here, $\langle\hat{a}_{S}\hat{a}_{I}\rangle$ is presumed to be real valued, which in general requires phase information in order to apply the appropriate quadrature rotation that maximizes the signal-idler correlation.

VI SNR of the coherent state illumination: heterodyne and homodyne measurements

To perform the coherent state illumination, we generate a coherent signal at room temperature and send it into the dilution refrigerator where the mode $\hat{a}_{S}$ is reflected at the JPC output port and passes through exactly the same measurement line as in the case of of the QI and CI protocols. The amplified signal mode is then used to probe the target region and measured via heterodyne detection. In the presence of the target the measured signal is given by Eq. (14) with $\langle\hat{a}_{S,1}^{\mathrm{det}}\rangle=\sqrt{\eta\,G_{S}}\langle\hat{a}_{S}\rangle$, and in the absence of target it is given by Eq. (15) with $\langle\hat{a}_{S,0}^{\mathrm{det}}\rangle=0$. Similar to QI, we perform data-processing on the recorded coherent-state outputs and use $M$ measurement results of the field quadrature operators $\hat{X}_{S}^{\mathrm{det}}$ and $\hat{P}_{S}^{\mathrm{det}}$ to perform a digital heterodyne detection with the following SNR

[TABLE]

For the digital homodyne detection we use phase information to rotate the signal to the relevant quadrature direction and obtain the improved SNR

[TABLE]