1. Introduction

Finding the optimal probe scheme for distant target detection with a high degree of accuracy is the main objective in the quantum radar domain. Related researches have been successful in improving the resolution and SNR of radar receivers [1–5], with quantum resources being demonstrated to possess better accuracy and noise resistance compared to conventional resources. In particular, the lower bound of the quantum target detection and positioning error decreases exponentially with the introduction of entanglement and squeezing [6–9], which mainly depends on the quantum dimensions of these non-classical resources [10,11]. Furthermore, the typical properties of quantum technology have been extensively explored in other fields, significantly promoting the development of quantum teleportation and quantum computation [12–14], as well as quantum cryptography and quantum simulation [15–17]. As a crucial branch of quantum radar, the QI scheme proposed by Lloyd [18] for detecting the presence of targets exploits the entanglement between signal photons and idler photons in the receiver, delivering remarkable advantages in SNR compared to conventional illuminations. The scheme features a simple application architecture and is widely compatible with various optical sources and reception modes [19–21]. Following that, pivotal improvements have been achieved in exploring more efficient optical sources and quantum measurement methods [22–24], with comprehensive performance analyses conducted in laboratory aspects [25–28]. Therefore, optical probing sources, atmospheric propagation, and quantum measurement theories have emerged as hotspots in recent research.

As the core of quantum radar research, substantial progress has been made to obtain maturer and brighter signal sources [29,30]. Particularly, SPDC technology was predicted in the 1960s to be capable of generating well-controlled twin photon states [31], mainly in the frequency and spatial domain, through the splitting process on precisely cut nonlinear crystals [32,33]. Burham et al. therewith demonstrated this widely used method in 1970 [34], which gets considerably matured in several nonlinear crystals such as $\textrm{Ba}{\textrm{B}_2}{\textrm{O}_4}$(BBO) [35], $\textrm{K}{\textrm{D}_2}\textrm{P}{\textrm{O}_4}$ (KDP) [36], and periodically poled $\textrm{KTiOP}{\textrm{O}_4}$ (PPKTP) in later research [37]. The SPDC process is categorized into a few types, for instance, Type-I and Type-II, according to the polarization relationship between the twin photons. To date, the photon pairs possessing different characteristics generated by the SPDC method have been steadily prepared in laboratory conditions [38,39]. As this technology matures in the field of quantum target detection, a photon statistics method based on the QI protocol has provided a novel way for the experimental realization of quantum illumination [40]. Moreover, they have analyzed the advantages in detail from both the SNR and error probability aspects. Cauchy-Schwarz parameters and the noise reduction factor (NRF) were calculated on the correlated thermal beams (THB) and twin beams (TWB) to quantify the quantum properties, which leads to significant conclusions [41,42]. In the experimental implementation of QI, photon numbers are recorded at each pixel via a CCD camera, and the noise is simulated by scattering a laser on Arecchi’s rotating ground glass. The signal and idler beams are going through different spatial paths, where the signal beam is directed towards the target area and the idler beam is transmitted directly to the radar receiver for the final correlation measurement. Thus, the presence or absence of the target can be discriminated by the differences in quantum statistical characteristics of the received photons.

Although various filters have been applied to probe distant areas with strong noise, it remains challenging to identify emitted photons and access the time of flight (TOF) information in complex optical environments. The surviving reflection signals (if the target is present) can easily become indistinguishable from noise, which significantly reduces the detection capability of the radar. Therefore, the quest for quantum skills that can replace conventional sources and receivers for elevating detection accuracy with the presence of strong noise and losses is of great significance. In current studies, the introduction of some methods based on twin-photon time correlation is gradually addressing these issues. Benefiting from the QI frame, the time-correlated property between twin-photon pairs is taken advantage of in Ref. [43], which nicely demonstrates the advantage of non-classical sources in target detection and imaging, from the perspective of SNR. Once the time precision of the measurement method is up to the picosecond level, it has been proven that the time-correlated twin photons can effectively filter out unknown noises and greatly improve the detection and ranging accuracy, with the time correlation advantages signified by estimation uncertainty and Fisher information [44]. Apart from the QI frame, high accuracy at the quantum level of ranging and imaging can also be achieved as long as the temporal and spatial correlations of the parametric down-conversion (PDC) photons can be precisely controlled [45]. Thereinto, PDC twin photons with intrinsic spatial correlation are emitted together to image the target, drawing an important conclusion that the simultaneous attenuation of twin photons in free space does not compromise the SNR gain. Generally, the twin photons generated by the pump photon are divided into different spatial paths and measured jointly at the receiver to identify their quantum state, which represents the mainstream of quantum detection currently [46]. However, at the receiver, Type-I twin photons cannot be separated for independent usages and require the CCD camera to match their arrival timings in multiple pixels, much less the fragile spatial correlation of twin photons during long-distance detection. Besides, false coincidences could significantly weaken the ranging and imaging capacity based on time correlation, as signal photons are likely to accidentally coincide with background noise photons or dark counts from the single-photon detectors (SPD) [47]. Together with the fragility of frequency entanglement, it is quite challenging to measure the surviving photons when only a few of them manage to reach the receiver.

While concentrating on the time correlations, optical paths of SPDC beams cannot ensure precise equal lengths due to some practical factors in the experiment, especially when only one of the beams is sent to detect targets (QI protocol), it is incapable of confirming the pairwise correlation of the two received photons. As a result, the temporal correlation between the two photons can only be measured by matching the time sequences recorded by SPDs [48]. It ultimately leads to a major expenditure of time for the continual step-matchings in the coincidence counting between the time sequences, to work out whether the time correlation exists inside them as well as the position of the target, which obviously violates the real-time principle of radar detection. Thus, significant accumulation in both time and data is necessary, plus the fact that each photon state can only be measured once, entailing enormous challenges for quantum storage and measurement.

In these previous explorations, twin photons without time alignment would prevent the receiver from identifying the time correlation in real-time, thereby increasing the complexity and requiring a significant amount of post-processing calculations, which is also the fundamental reason for the practical implementation remaining immature. Nevertheless, the time alignment twin-beam (TAB) method proposed by us holds important research significance and can address these challenges perfectly. The characteristic of HOM highly fits the time alignment requirement of twin photons, which also provides a compact and stable spatial construction for the output. The TAB scheme enables us to promptly reconstruct the probing waveform which has been destroyed during detection while eliminating most of the receiving noise, by simply performing a single real-time coincidence counting. It can be expected that, under the same coincidence window width and energy level, the simultaneous probing of twin beams in TAB may result in higher losses for the final coincidence counting, bringing out slightly less SNR than the QI scheme. Despite this fact, we have demonstrated that the TAB scheme still maintains stable SNR advantages over conventional illumination. More importantly, the SRCC and time alignment method via the HOM interferometer permit a shorter coincidence window which reaches the femtosecond level to reap an explicit SNR advantage against the QI scheme under the same energy. Because for the QI radar, the long-distance probing requires so many times for matching coincidence countings that the coincidence window and matching steps at the femtosecond level will lead to enormous calculation amounts lasting days, which are unattainable in real radar detections, by contrast, the proposed TAB method only perform coincidence counting once. However, the pity is that the detector dead time is usually at the nanosecond level, with the smallest jitter of existing SPD technology hitting about $8\,{ps}$ only, which will destroy the femtosecond-level time alignment between twin photons and lower the receiving efficiency. Thus in our femtosecond TAB experimental design, the circuit jitters in the receiving system are assumed to be less than the coincidence window from affecting the counting result, hereinto the arrival time sequences of receiving photons are composed of simulated time-alignment twin photons and real noise photons, apart from that, the whole CI and QI schemes are completed in actual experiments. In addition, we have also verified that twin photons tolerate well the majority of target surface fluctuations by applying coincidence windows in various lengths, providing stable coincidence counting results. Based on these results, the proposed TAB method enables real-time measurement abilities of time-correlated quantum peculiarity at the receiver and has good robustness for all kinds of targets and noise environments. More importantly, the corresponding receiver has a simple structure and can be easily implemented in practical experiments, while the higher SNR and better signal fidelity have also paved the way for the experimental implementation of quantum radar.

The experiment establishes a QI control group and a CI control group. The illumination source in the QI group is obtained directly by the Type-II SPDC in our laboratory, i.e., the SPDC twin beams without time alignment, and the TAB scheme is simulated according to the photon intensities of the QI scheme, to highlight that the shorter coincidence window benefited by the HOM time alignment and SRCC not only can achieve the real-time detection of quantum information but also performs a better SNR than QI scheme. Correspondingly, the CI group is designed to directly illuminate the target with one of the SPDC beams of QI while the other beam is blocked, to reflect that the temporal correlation of the TAB method possesses a similar capacity as QI in reducing unexpected noises and acquiring extremely high SNR gain. The results obtained in the above control groups also demonstrate the viability of double losses ${\eta ^2}$ brought by the simultaneous propagation of the twin photons in the TAB scheme. To compare the performance of different illumination schemes in our experiment, we define the SNR value by calculating the average power over four pulses time from the perspective of photon counting and coincidence counting. Since the SNR in the unknown background typically represents the target detection capability of radar, we show the received waveforms of TAB and the CI group in the good noise region as well as the bad region, and provide the SNR results of the three protocols along with different noise backgrounds.

Moreover, our TAB protocol operated on the frame of optical pulse radar successfully explores the time alignment properties of twin beams to reduce the disturbances from environmental noise and make the quantum properties measurable in real time. As it is well known, the twin photons in SPDC are generated simultaneously, thereby making temporal alignment between the photon sequences possible in the experiment. Due to the transmission properties of twin beams in nonlinear crystals and the unknown phase differences from the SPDC process, temporal mismatch of the twin beams is inevitable [49], while the HOM effect [36] offers a solution to precisely restore the alignment and these probe units can be captured at the interferometer outputs. Furthermore, to ensure the probe source can be effectively detected, the optical structure should be adjusted to make the twin photons out of the HOM interferometer maintain parallel as much as possible in free space. Because the probing facula is small enough, the down-converted twin photons can be approximated as a bigger detection unit and share the same scattering characteristics.

In the transmitter and receiver redesigned in our experiments, frequency-degenerated twin photons produced by the type-II SPDC process are selected to meet the HOM conditions. After the HOM alignment, twin photons will maintain a fixed and negligible time interval (caused by the phase difference between transmission and reflection). It is worth mentioning that these SPDC twin photons are entangled in polarization, one of them is horizontal and the other is vertical. That is the key to analyzing them at the receiver: the received photons filtered through the optical layer are divided into orthogonal polarization channels by a polarization beam splitter (PBS) and recorded by two SPDs respectively before the coincidence counting circuit. Another major differentiator is the SRCC method carried out on a series of short-enough time slices at the receiver, and the counting values of the time slices are shown in the histogram to reconstruct the signal waveform. Even though the phase disturbances from the propagation and scattering process will bring optical path differences into TAB, which unavoidably have negative influences on the final coincidence counting. However, due to the time alignment process, these time mismatch at the receiver is so small that allows us to carry out the SRCC in an extremely short time window (femtosecond level) without step-matching, to be more specific, it always corresponds to the peak of the second-order correlation function of the coincidence count, making the stable temporal alignment of the twin beams necessarily required. Our work brings a different detection and measurement method that can improve the SNR performance of illumination and reconstruct the signal waveform from terrible noise background via the time correlation of SPDC twin photons, and its advantages are analyzed mathematically and proved experimentally in detail.

The detection system of the TAB illumination is exhibited in illustrations, the theoretical feasibilities of the time-alignment and SRCC are also demonstrated in the following sections. To introduce the HOM effect, quantum analyses of the Beam Splitter (BS) with incident photons at both input ports are taken into consideration to provide the theoretical foundation for recovering the time alignment. After that, the detection method and the signal reconstruction performance of the TAB in pulse radar are analyzed in theoretical derivations and shown in actual experiments as well as the femtosecond window simulation results, compared with the actual CI radar and QI radar under the same experimental setup. To adapt to various target surface fluctuations during detection, the effect of the coincidence window size on the illumination SNR is also considered, and the result shows stable SNR advantages of TAB with all the regular windows. The next part primarily illustrates that the TAB protocol offers a stable SNR advantage compared to QI and CI protocols at any level of noise intensity, and the influence of noise intensity on final coincidence count variance is also given. As for the technical specification of SPD, the precision and resolution have continued to improve, and the femtosecond level coincidence window will be applied to practice ultimately with the development of circuit technology, before that, the picosecond level TAB still possesses stable SNR advantages near the QI and at least the real-time virtue. Meanwhile, the practical parameters of the illuminator and the coincidence counting circuit are given in detail as well.

2. Methods

2.1 Temporal alignment and illumination transmitter

In the experimental design of the illumination source, polarization-entangled light is considered as a suitable detection source for easy preparation and satisfying the requirements of subsequent operations. When a single mode laser (pump light) with frequency ${\omega _\textrm{p}}$ illuminates a 2 mm BBO crystal that is cut for the Type-II SPDC, a small part of the incident photons can split into lower-frequency photons with frequencies ${\omega _\textrm{s}}$ and ${\omega _\textrm{i}}$, the “$\textrm{s}$” and “$\textrm{i}$” above represent “signal” and “idler” photons respectively [33]. After selecting two specific spatial positions, we can obtain a steady production of polarization-entangled twin photons, which are non-collinear obviously and have orthogonal polarizations. Thereinto, due to the particular character of the Type-II SPDC, the frequency-degenerated two-photon in which ${\omega _\textrm{s}} = {\omega _\textrm{i}}$ are emitted in two circular cones side-by-side with orthogonal polarizations ${{|H \rangle } / {|V \rangle }}$, and these two circular cones intersect at two points a and b in the near field, as shown in Fig. 1(a, b). Moreover, photons collected at these two points are unable to be distinguished for which cone they belong to, such that we can get a stable polarization-entangled twin beams state $|\varphi \rangle $ [35]:

 

Fig. 1. Graphical descriptions of polarization-entangled SPDC process and photon paths in HOM interferometer.

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To better understand the internal properties of the above polarization-entangled states while passing through the HOM interferometer, it is necessary to analyze the output of the HOM interferometer at the single photon level. When the down-converted twin beams fall into a 50:50 BS from both input ports as shown in Fig. 1(c, d), there are two possible cases: (1) Two photons arrive at the BS one after another from two ports, with their time interval larger than their coherent time. Then, photons that go through or get reflected with equal probabilities behave as independent individuals in the whole process, and the coincidence counting value occurs between the outputs. (2) Simultaneous incident of twin photons: the position of BS is adjusted to make two photons arrive at the BS with their time interval near zero, thus the input state can be expressed as ${1 / {\sqrt 2 }}({{{|H \rangle }_0}{{|V \rangle }_1} + {{|V \rangle }_0}{{|H \rangle }_1}} )$ (input port ${\hat{\alpha }_0}$ and ${\hat{\alpha }_1}$) for this situation. Different from case (1), the arrived twin photons blend together and come out through either of the ports (output port ${\hat{\alpha }_2}$ or ${\hat{\alpha }_3}$) with equal possibilities, hereinto, one photon goes through and the other must be reflected. Then the output state of the HOM interferometer can be written as:

In the just described case (2) with twin photons simultaneously incident, the output state ${|H \rangle _2}{|V \rangle _3}$ or ${|V \rangle _2}{|H \rangle _3}$ does not exist can be interpreted as the destructive interference of two possible output situations: both transmission or both reflection. This requires that the two output states are indistinguishable, that is, it can not be measured which one is reflected or transmitted. Obviously, when the input state is simply PDC photon pairs ${|H \rangle _0}{|V \rangle _1}$ or ${|H \rangle _1}{|V \rangle _0}$, the measurement of polarization can directly distinguish between the two output cases. Due to the polarization-entanglement properties in (1), the polarization of the input photons at ports ${\hat{\alpha }_0}$ and ${\hat{\alpha }_1}$ cannot be determined unless we perform destructive measurements to make them collapse into normal unentangled PDC states. Therefore, the two outputs $|{{\varphi_{\textrm{BS}}}} \rangle$ generated by the two path cases are indistinguishable, as is shown in Fig. 2. It can be concluded that the frequency degenerate twin photons with polarization entanglement can also cause the HOM effect, and no extra adjustments to the optical path are required in the experiment.

 

Fig. 2. Destructive interference of two path cases.

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In addition to the HOM alignment method mentioned above, a more acceptable method, which can make normal unentangled PDC states with opposite polarizations also applicable to our TAB scheme, is shown in Fig. 3. That is, by placing a half-wave plate (in the dotted box) to rotate the polarization of one beam by 90 degrees, to eliminate the polarization orthogonality between the twin beams, so that the HOM effect will be more intuitive, as described in the Ref. [36]. Since the function of the whole HOM interferometer is to align the optical paths precisely to guarantee the simultaneous arrival of the twin photons, the half-wave plate should be removed after the optical path alignment is completed to restore their polarization orthogonality, which can be considered as the pre-alignment of the twin beams. By adjusting the width of the coincidence counting window, it can be drawn that the optical path difference introduced by the half-wave plate does not affect the result of the ultimate coincidence counting, thus the optical system after this pre-alignment via the temporary half-wave plate could meet the measurement conditions of SRCC and supplement more unentangled sources to select.

 

Fig. 3. HOM alignment after the disentanglement of twin photons.

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In principle, neither the entanglement in frequency nor polarization between two-photon is necessary for the time alignment process. What should be emphasized in this scheme is the importance of temporal correlation of the twin photons, hence in the transmitter part, simply parametric down-conversion photon pairs can meet the requirement in the same way. However, the target detection system introduced in the present paper considers how to separate the two-photon pairs at the receiver so as to comply with the subsequent coincidence counting operation. Therefore, also for simplicity of optical setup, the polarization-entangled two-photon with stable polarization characteristics in the atmosphere is undoubtedly the best choice (its HOM feasibility has been demonstrated in the previous section). More importantly, the possible reflected photons can be separated into orthogonal polarization channels with PBS at the receiving end, which is crucial for the final signal processing.

The precise temporal relationship between the interior of the twin photons should be carefully considered before performing precision HOM alignment, hence the “correlation case” (2) with twin photons simultaneously incident is discussed emphatically as follows. In the practical experiment, the material of 50:50 BS is used to be single-layer dielectric, and then an extra phase difference of ${\pi / 2}$ always occurs between the reflecting and transmitting of incident twin photons, i.e. an invariable skewing $\exp ({\pm} i{\pi / 2}) ={\pm} i$. Thus at the outputs, a constant ${\pi / 2}$ phase difference can not be eliminated between the process of transmitting and reflecting photons (see Fig. 4).

 

Fig. 4. The phase relationship between twin photons’ reflection and transmission.

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In the optical setup for the HOM temporal alignment, a 32 mW laser with a stable output at 405 nm wavelength is utilized, hereinto, an optical pulse chopper is used to convert the continuous light into optical pulses. This pulse source generates two frequency-degenerate and polarization-entangled beams via SPDC in a 2 mm thick block of BBO crystal with specific spatial modes selected. Such twin beams have been prepared for the time alignment, which will be sent into a 50:50 BS for the next step of HOM interference, and beams must be well overlapped on the BS for a noticeable coincidence count dip. As previously analyzed, as long as the position of the BS is accurately adjusted, the two SPDs connected to the BS output will exhibit a clear coincidence count dip, indicating that the two down-converted photons are emitted from only one output of the BS and are aligned in time. As discussed before, the aligned twin photons maintain a fixed difference ${\pi / {2{\omega _\textrm{s}}}}$ inside, corresponding to the time delay of 0.675fs in our experimental settings, which is too short to be detected by the current SPD technology. Consequently, twin beams from different paths are precisely aligned in time and properly prepared for the detection emission, the optical path structure is shown in Fig. 5(c) as well. Though the design and experiment setup of the TAB transmitter has been completed in our experiment, we should add that the practical time-alignment photons have to be replaced by computer simulations instead, due to the destructive reception for high precision time-alignment of current SPDs. Furthermore, the polarization of light can be well-maintained during its transmission and scattering in free space over long distances, which lays the foundation for subsequent coincidence counting operations.

 

Fig. 5. Transmitter design of the conventional and time alignment illumination. (a) Conventional transmitter with classical light from one of the SPDC beams. (b). The QI transmitter with the same energy as the TAB transmitter. (c) Time alignment process through the HOM interferometer. During time alignment, the position of the BS is adjusted appropriately to ensure that the SPDC photons can arrive at the BS simultaneously.

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Due to the small enough illumination spot (one of the HOM outputs) during detection, the twin photons always get reflected at the same interact point of the target simultaneously. Therefore, it can be approximated that the two photons experience the same scattering and reflection events during their propagation, and their spatial structure remains unchanged. In the CI control group of the experiment, the twin photons are replaced by classical photons, obtained by one arm of the TAB, and the pump energy remains the same (see Fig. 5(a)). Similarly, to demonstrate the advantages of the TAB protocol over the optimal QI radar at the same energy level, we establish and actually implement the QI group by replacing the illuminating source with the non-aligned SPDC beams which are identical to the beams before the HOM alignment in the TAB protocol, the transmitter designs are shown in Fig. 5(b),(c).

2.2 Conventional and time alignment illumination receiver

If a target is detected by the CI radar, a sequence of periodic pulses emitted by the radar transmitter will be reflected to the receiver, otherwise, only background photons can be received. The SPD placed at the input of the conventional laser radar receiver is utilized to record the incident photon count and their arrival time sequences. The number of pump photons is denoted by ${N_p}$ and the number of noise photons received is ${N_b}$. When the target is present, the total number of photons received at the receiver is:

Clearly, if the target is absent, ${N_t} \equiv {N_b}$ and ${f_c} \equiv{-} \infty$. When the target is present, the CI detection method and the corresponding received waveform are shown in Fig. 6(a). It can be observed that the pulse intensity of the probing signal is significantly higher than the intensity of the background noise, which only occurs under ideal conditions, i.e. a near-field environment with strong signals and weak noises, we refer to this as a good region. However, as the detection distance increases, the probing photons will suffer greater losses and be drowned out by noises, making it impossible to extract useful information from the received photons, we thus refer to this as a bad region. To summarize, conventional illumination can only utilize the intensity and time information of optical pulses therefore, increasing the output power of the laser is a common approach to improve the target detection SNR. However, even without regard to the power limitation of the radar transmitter, some high-brightness light sources remain insufficient in compensating for high losses and strong noise.

 

Fig. 6. Schematic diagram of the radar detection system. (a). Target detection in conventional pulse radar. (b). Target detection in TAB pulse radar. The waveforms in green lines correspond to two polarization channels and the result of SRCC is exhibited in purple.

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Before introducing the target detection method based on the coincidence counting algorithm, it is necessary to conduct a standard quantum analysis of the coincidence counts of the polarization-entangled twin beams. We consider that signal and idler fields are detected through small apertures at fixed positions far enough from the illumination source, thus only one spatial mode of photons is chosen by each detector. For the sake of generality, we can ignore the polarization entanglement of the twin photons when calculating the coincidence count value, as it only takes effect in the dual-channel processing at the receiver. Hence we take the analyses under the frequency state $|\omega \rangle $, the emitted quantum state Eq. (1) produced at moment t takes a very simple form:

After the action of the operators on the quantum state and integration over ${\omega _1}$ and ${\omega _2}$, the coincidence counting rate is written as:

Therefore, the final SNR of the TAB scheme in n slices can be written as:

At the TAB radar receiver, polarization entanglement between the twin beams can be utilized, together with PBS and optical components covered with 810 nm filtering coating, to split the received photons into horizontal and vertical polarization channels while reducing part of the background noise count. The increase in detection distance and background photons in the bad region will result in a significantly lower detection photon intensity compared to the noise photon intensity, ultimately rendering the target waveform invisible. The photon intensity of the two polarization channels in the temporal domain is shown as the green curve in Fig. 6(b). Unlike photon counting in conventional illumination, we innovatively use the SRCC value between the twin beams to reconstruct the probing pulses. In order to retain the distance information of the target, we perform SRCC with a fixed coincidence window for the twin streams split by the PBS in each time slice, without the necessity to balance the delay (which is unknowable in the laboratory). Therefore, the coincidence counting circuit can rapidly calculate the counting value of each slice, and the histograms of all time slices form the reconstructed pulse waveforms in the temporal domain. The eventual real-time pulse waveform obtained by coincidence counting values between the dual channels is shown as the purple curve in Fig. 6(b). The noise situation of both illumination protocols in Fig. 6 is set to a good region, this allows us to observe the detected pulses distinctly.

3. Results

3.1 Signal reconstruction of time alignment illumination

Reconstruction experiments are set under the actual conditions that the intensity of signal or idler photons is ${N_S} = 1000k\;cps$, and the width of pulses is set to be $T = 1\;ms$ by adjusting the rotational speed of the optical chopper. The width of the counting window that we have selected is ${T_c} = 160\;ps$ while the deadtime of the SPD is about $70\;ns$. The optical components used in the receiver are coated with a filtering layer with a central wavelength of $810\;nm$, together with an IF to filter out part of the background light. The $\eta $ above including both the transfer and reception efficiency is $\eta = 0.36$ in our experiment, hence the overall efficiency of the TAB protocol would be ${\eta _t} = {\eta ^2} = 0.13$. Noises in each channel are generated by a noise laser producer, which also stays at the 810 nm wavelength and can be flexibly adjusted to simulate noise background in different intensities.

Firstly, in the good region where the average photon number of noisy photons is significantly lower than that of probing photons, i.e. $\left\langle {{N_b}} \right\rangle \ll \left\langle {{N_s}} \right\rangle $. The obtained pulse waveform in the CI protocol is shown in Fig. 7(a). In both illumination protocols, the intensity of detected photons is much higher than the energy of noise photons, thus providing clear pulse waveforms and containing arrival time information. As such, together with the great photon consumption in the inefficient SPDC process and ${\eta ^2}$ probing losses, the TAB waveform peaks have underperformed the CI and QI ones, similarly in terms of the receiver SNR. Even so, as the post-coincidence waveforms shown in Fig. 7(c), only a tiny amount of noise fake counts are left, which illustrates that the SRCC does have played a role in noise filtering from Fig. 7(b) to Fig. 7(c), albeit not quite as pronounced because of the low-interference environment. From the perspective of radar detection, the TAB receiver still maintains a good SNR which enables checking out of the target echo signals simply by a noise threshold, even filtering out a pure signal space in real-time under this ideal low-interference situation.

 

Fig. 7. The received waveform of conventional and time alignment radar. (a-c). Radar in the good regime $\left\langle {{N_b}} \right\rangle \ll \left\langle {{N_s}} \right\rangle $. (d-f). Radar in the bad regime $\left\langle {{N_b}} \right\rangle \approx 20\left\langle {{N_s}} \right\rangle $. (a, d). Waveform obtained by CI receiver. (b, e). Dual-channel waveform obtained by TAB receiver. (c, f) Reconstructed waveform through dual-channel coincidence counting, the amplitudes represent the coincidence counting values in each $10\;\mu s$ time slice. To make it easy to compare the results, abscissa in (a-f) reports the time in ${10^{ - 5}}s$ bins.

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After the investigation of low noise situation, the bad region with high-level noise $\left\langle {{N_b}} \right\rangle \approx 20\left\langle {{N_s}} \right\rangle $ is also analyzed, which has wider applications and higher research value in realistic scenarios. To make the results more representative, the noise laser intensity is increased to twenty times that of the probing signal in each channel to have the pulse waveform ruined completely, while the signal intensity remains unchanged. By doing so, the pulse waveform received in each polarization channel turns invisible, making it impossible to proceed with further signal processing operations. The CI waveform of the received pulse under these strong noise conditions is shown in Fig. 7(d), similarly, it can be inferred that the TAB waveform received in both channels would completely disappear as well, as shown in Fig. 7(e). In this situation, neither the presence nor the arrival time of the pulses can be figured out from the signal intensity, which is mainly attributed to the high background noise and significant signal attenuation caused by the long detection distance. Nevertheless, time alignment between the twin photons is not sensitive to strong noise and signal attenuation, no matter what noise intensity it is, TAB photons will always be promptly matched and accurately counted through real-time coincidence filtering. In the bad region shown in Fig. 7(d-f), the coincidence window width is set to remain ${T_{TAB}} = 160\;ps$ in signal processing, to demonstrate the actual feasibility of the TAB scheme for real-time noise filtering and signal reconstruction. Extending this window would result in fake coincidence counts that are caused by false coincidences among signal photons, background photons, and dark counts, which can be considered as a new-type noise within the coincidence counting result. Ultimately, by employing the SRCC method, we can obtain a reconstructed detection waveform that is clear and accurate, as shown in Fig. 7(f). It can be seen from the peaks in panels (c,f) that the reflected twin beams received in the TAB would suffer different consumption separately, directly leading to the post-coincidence peaks relatively below the CI signal and that of QI, even so, the sacrifice of coincidence intensity is exchanged for the filtering of nearly all the noises as well as the real-time advantage. Due to precise time alignment and the SRCC method, we can employ the coincidence counting window as short as possible to filter out unwanted noise, which is up to ${T_{TAB}} = 16\;fs$ in our following simulations, just this feasibility of the femtosecond window endows the TAB with an enormous potential for exceeding the SNR advantage of QI scheme. What needs illustration is that the circuit technology limits, such as time jitter $\Delta \tau$ in Eq. (6) and the dead time of SPDs, are out of consideration for the TAB simulations in our next section, because our experiment is aimed at demonstrating the advantages of quantum properties brought by the TAB scheme, while the circuit time jitter which is about several picoseconds at present will reach the femtosecond level ultimately.

3.2 SNR of time-correlated illumination and conventional illumination

In the previous sections, we validate the feasibility of reconstructing the pulse waveform by coincidence counting method when the coincidence window is set to $160ps$, which corresponds to a target surface fluctuation depth of $24mm$. Propagation in free space can easily bring unknown optical path differences inside twin photons, which are typically very small since they are primarily determined by the reflection process on the target surface. The tiny transverse distance between the probing twin photons only leads to extremely short path differences, as the twin photons almost get reflected at the same point on the target surface. However, considering the randomness of the target surface in experiments, the effects of coincidence window width on SNR are studied from femtosecond to nanosecond, which covers the time delay caused by most of the target surfaces (with a time difference of approximately $6ps$ for each $1mm$ of depth). Also for the sake of generality, we investigate the effects of various coincidence window widths on illumination performance by capturing signals during 4-period pulses and calculating the SNR value [Eq. (4) and (11)]. Simulation results shown in Fig. 8(a) exhibit great robustness of TAB to any coincidence windows, which is applicable for the vast majority of coincidence events. It has been confirmed that widening the coincidence window appropriately when the target surface depth is at the order of meters does not introduce notable false countings, and the noise filtering effect is basically the same as that shown in Fig. 7(f). The TAB result in Fig. 8(a) also applies to QI protocol, since they both benefit from the time-correlated properties, only when the coincidence window is overly widened does the SNR of time-correlated decrease to the level of CI protocol.

 

Fig. 8. SNR performance of the time-correlated and conventional radar. (a). SNR advantages in function of the width of coincidence counting window. The mean number of noise photons is about $20Mcps$, and the mean number of signal photons is about $4Mcps$. (b). The variation of SNR values along with the mean number of noise photons. The width of the QI window is set to $160ps$, while the TAB window is set to $16fs$, and the mean number of signal photons is about $4Mcps$ in both the TAB and QI scheme. The red and blue marks refer to the SNR value of the TAB and QI radar, respectively, while the green marks denote the SNR value of the conventional radar. Variances of time-correlated (TAB and QI) protocols affected by the coincidence window width and background noise are also analyzed, respectively, which is represented by the uncertainty bars in dash lines.

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In Fig. 8(a), the SNR of conventional illumination is constant and unaffected by the coincidence window width, since the CI SNR is totally decided by the energy intensity. Moreover, false counting hardly exists when the width of the coincidence window is up to the femtosecond or picosecond level, which enables TAB detection events to reach great and stable SNR advantages while almost filtering out all the noises during the signal reconstruction process until the window width comes to the nanosecond level. TAB and QI protocols are both kinds of time-correlated illumination, with their coincidence window widths differing at the receiver, hereinto, the former corresponds to the femtosecond level curves, while the latter does not have realistic conditions for femtosecond windows in so many times delay-balancing that can only be closely approximated by the picosecond level curves in Fig. 8(a). Several experiments are finished to calculate the stability of time-correlated illumination SNR, the variance represented by the dash lines in Fig. 8(a) shows that the SNR uncertainty gets larger while the width of coincidence windows increases, which primarily results from the coincidence events between signal and background photons.

The detection environment of radar should be considered as an apriori unknown background, where the noises are incapable of being eliminated simply by a reference threshold of photon numbers received, compared with the possible reflected probing photons when the target is present. In particular, the intensity of the photon countings in conventional illumination is not informative regarding the presence/absence of the object when the background noise is very strong. We underline that this unknown background accounts for a “realistic” scenario where the noise distribution properties can randomly change and drift with time and space. Moreover, all possible noise fluctuations should be taken into account to make the proposed method closer to the “realistic” scenario, thus we analyze the SNR performance of the TAB and QI detection method based on the coincidence counting over the conventional method, along with the increase of the mean value of background noises and the result is shown in Fig. 8(b). In the QI counter group, we realize the target detection by matching the reflected and idler photon time sequences with the step size being $100\textrm{ps}$, and the SNR value based on the peak of coincidence counting value during the matching process is calculated similarly as the TAB scheme, the results are shown in Fig. 8(b) in blue. It can be seen that the advantages of SNR in the TAB scheme are huge and fully demonstrate the strong robustness against noises while the SNR of conventional illumination is already quite small in that bad regime. Due to the femtosecond potential of the coincidence window benefited by the time-alignment and SRCC methods, SNR in the TAB scheme holds stable advantages against that of the QI method, let alone the CI scheme, even though twin beams in the TAB suffer double losses ${\eta ^2}$. Also for the sake of practical realization, TAB and QI schemes have been tested with the same coincidence window ${T_{TAB}} = {T_{QI}} = 160\;ps$, as mentioned before, it is the double losses ${\eta ^2} = 0.13$ that have sacrificed part of the SNR benefit of TAB in exchange for real-time detection while still keeping up its remarkable advantage upon the CI scheme, the consequences of that can simply be derived from the theoretical part in previous sections so we no longer show them in figures. Likewise, variances of the SNR value affected by the mean number of background photons ($20 \sim 500Mcps$) hold steady all the time, except the first half of the TAB variance, its comparatively large uncertainty is mainly because the noise intensity is low and the TAB coincidence window is so short that only very few noise countings are obtained. It’s worth noting that noise coincidence values that are several times larger than that of the signals may occur suddenly in some time slices, which will cause the TAB scheme SNR to decrease faster along with the increase of $\left\langle {{N_b}} \right\rangle $, to avoid this situation, a threshold with a value of the theoretical maximum of the signal is applied in our signal processing part to filter out those outliers.

4. Discussion

The time alignment illumination designed in our work is based on the Type-II SPDC process on nonlinear crystals, while the active brightness of the down-converted lights is the most concerned point in detection events. In view of the low SPDC efficiency (about $1 \times {10^{ - 8}}$), only very few available photons in frequency ${\omega _\textrm{s}} = {\omega _\textrm{i}} = {\omega _\textrm{p}}/2$ will be captured, particularly after the spatial selection of the phase-matching conditions via the apertures and IFs placed at the well-adjusted spatial positions. In realistic situations, however, the inherent frequency spreads make it incapable of guaranteeing the single-mode frequency of the selected photons, partly on account of the spatial size of the apertures and the optical efficiency of the filtering layer. Therefore, the down-converted photon state should be revised as $\int {d\omega \psi (\omega )|{{{{\omega_\textrm{p}}} / 2}\textrm{ + }\omega } \rangle } |{{{{\omega_\textrm{p}}} / 2} - \omega } \rangle$, where the $\psi (\omega )$ corresponds to the weight function which is peaked at $\omega \textrm{ = }0$. Moreover, the coincidence counting value after the HOM interferometer should have decreased to zero once we complete the time-alignment of the twin beams, however, the spatial overlap of the faculae can not be precise enough so that the coincidence counting value becomes quite low but always larger than zero. So far, all the influences above have merely reduced the available time-correlated photon pairs that we ultimately utilize for probing. Although the spatial correlation of SPDC beams has been verified in relevant research [50,51], subsequent beamforming of the transmitter requires manual adjustment of the optical geometry in the HOM interferometer to ensure that the twin beams are emitted as parallel as possible, so as to guarantee their optical-path differences acceptable and can be received effectively at the receiver [52]. The reason why we utilize the Type-II SPDC is that we can separate the signal and idler photons into different polarization channels for subsequent coincidence counting at the receiver, no matter which polarization states they belong to. The simultaneity of the twin photons produced by the SPDC process is the key to our proposed method, which makes it possible for the precise and steady time alignment of twin beams. The primary distinction between the TAB pulse radar proposed in this research and the conventional one is that we can focus on the two-photon simultaneous detection in one probe unit, and successfully unscramble the receiving unknown photons from strong losses and noise to obtain the hidden time-correlated information in real time.

5. Conclusion

In this research, the feasibility analysis of the time alignment based on the HOM interferometer has been completed, as well as the demonstration of real-time signal reconstruction, making them the major techniques to upgrade the QI protocol to higher SNR performance and the realization of real-time detection. It can be observed from the SNR results that the TAB properties hold steady advantages and effectively overcome the bad effects of the inevitable noise and losses in the whole radar process. In the method section, we provide detailed illustrations and analyses of the dual-channel photon receiving and target detection method based on the SRCC algorithm, and the ultimate experimental results consistent with the theoretical derivations are obtained. Thereinto, theoretical analyses of the SNR values in the CI and TAB scheme are finished in detail, and the explicit improvements in terms of experimental settings are demonstrated under the apriori unknown background and various lengths of coincidence counting window. For the TAB scheme we propose, contrasts with the conventional method have proved to be stark, and the biggest improvement over the QI protocol is the first realization of real-time detection of quantum information while similar SNR enhancement is obtained as well. The time correlation of twin photons after alignment can greatly reduce the noise level in the good regime and perfectly reconstruct the detection signal in the bad region, even suffering from ${\eta ^2}$ losses. As a result, elements of valid reality such as low brightness sources, poor efficiency of down-conversion, long detection range, and strong noise environment are getting tolerable by virtual of this excellent SNR gain. In conclusion, our proposed quantum radar scheme including the whole new transmitter and receiver can greatly enhance the capacities of target detection in illumination events and is widely appropriate for every possible target feature. In addition, this probing and measurement method of time alignment also provides new insights into quantum ranging, imaging, and quantum communication, which can greatly reduce the system time accumulation and obtain better position precision.