Abstract
1. INTRODUCTION
Quantum metrology provides a route to conquer the measurement sensitivity limitation imposed by quantum noise in sensing devices [1–3]. As a unique quantum resource, squeezed and entangled states present the feature of quantum correlation, which can suppress statistical scaling of errors in interferometric measurements, and thereby enhance measurement precision, until approaching and reaching the Heisenberg bound [4–6]. It holds particular relevance to the power constraint introduced by optical damage or quantum measurement back-action, and an ultra-weak signal that is thoroughly submerged in inherent quantum noise. Laser radar is a prime case of ultra-weak signal sensing, and has an urgent demand for measurement sensitivity beyond the shot noise limit [7,8].
Since quantum metrology was recognized in the 1980s [9], both the theoretical scheme and experimental technology of quantum metrology have advanced dramatically [10,11]. In this regard, numerous demonstrations of quantum-enhanced metrology have been reported, for example, to enhance the performance of gravitational wave detection [12–15], magnetic field detection [16,17], and biological measurement [18] and imaging [19,20], and are expected to be applied in more practical areas. Furthermore, quantum sensing has been upgraded from a single sensor to a sensor network to realize spatially distributed sensing of multi-parameters. Theoretical schemes of distributed quantum sensing have been proposed in both the discrete variable (DV) domain [21–24] and continuous variable (CV) domain [25–29]. By utilizing entangled photons [30,31] or entangled states [32], the average optical phase was estimated experimentally with precision beyond what is achievable with seperable probes. Very recently, Xia
In this paper, we experimentally demonstrate a quantum sensor network empowered by four-partite CV entanglement, optimizing the quantum advantage in global parameter estimation including the higher order derivative of radio-frequency signal phase. The global parameter of four sensors , where is the weights for different sensing problems, is estimated in our experiment, achieving quantum noise reduction of . The quantum sensor network is configured with an equally weighted distribution to maximize the quantum advantage for measuring the angle of arrival of the return signal along three independent dimensions, showing quantum positioning. The sensor network is reconfigured with a weighted distribution of :3::1/3 (2::4:) to maximize the quantum advantage of the first (second) derivative of PAR phase, demonstrating quantum ranging.
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PAR consists of multiple sensing elements appropriately arranged in space, whose amplitude and phase can be independently controlled to provide full space scanning without the influence of receiving sensitivity [38]. Space positioning can be realized by measuring the angle of arrival of the return signal in three dimensions [39], which can be inferred from the phase difference between two sensor nodes of each dimension [33]. Thus, to achieve space positioning, it requires at least four sensor nodes. As the other key capacity, target ranging, dependent on the angle of arrival and its derivative of the return signal in one dimension, can be obtained by configuring a diverse weighted distribution among these sensor nodes. As such, a reconfigurable beam splitter network (BSN), in terms of minimum estimation variance, distributes the needed portion of a squeezed state to these sensor nodes, leading to the optimal quantum advantage in different cases, which serves as the estimation of the angle of arrival and its first (second) derivative beyond the shot noise limit. There are two cases of sensor network topology: equilateral star topology for space positioning, and linear topology for target ranging. Compared with the demonstration of only an average outcome among all of sensor nodes, the richer physics and more intriguing application scenario of a distributed quantum sensor network are unveiled by our work.
2. QUANTUM POSITIONING PROTOCOL
In the quantum positioning case, three sensor nodes are arranged in an equilateral star topology, while the central node is situated at the triangle center; the schematic illustration of quantum positioning is represented in Fig. 1. When the target occurs, we can acquire three independent angles of arrival in three-dimensional space, and the target is positioned at the intersectional region of three angles of arrival. Critically, the distributed quantum sensor network must be optimized to generate minimum outcome uncertainty in a given distributed sensing problem. Here, to meet the requirement for simultaneous estimation of three angles of arrival with minimum positioning uncertainty, dependent on the phase difference between two sensor nodes with the function of , a CV quantum sensor network with an equally weighted distribution is constructed. Under the circumstances, the additional noise coupling vanishes, leading to the minimum estimation variance of the PAR phase that depends on the initial squeezing factor. The equally weighted distribution is experimentally verified when the average variance for PAR phase reaches minimum. At this point, the uncertainties of three phase differences , , and represent the best compromise, and will have a boosted performance for target positioning.
Figure 1.Schematic illustration of quantum positioning based on CV entangled network. (a) Principle of quantum positioning. (b) Structural diagram of PAR network.
A schematic illustration of our experimental setup is represented in Fig. 2. The squeezed state of light is generated by a below-threshold optical parametric amplifier (OPA), which has been shown in our previous publication in details [40]. The only difference is that the relative phase between the seed and pump beam is controlled to zero to ensure our OPA operates in parametric amplification condition, generating the quadrature phase squeezed state [; is the squeezing factor]. The BSN, which consists of three pairs of half-wave plates and polarization beam splitters, can be set to any splitting ratio by tuning the angle of half-wave plates. After the BSN, four partite CV entangled states with correlation variance [; is the phase quadrature of each partite] are generated and distributed to four sensor nodes. Each sensor node entails an electro-optical modulator (EOM) to simulate the return signal from the target, and homodyne detection (HOM) to discern the return signal with sensitivity beyond the shot noise limit. In each node, the relative phase between the local oscillator and sensing beam is controlled to by utilizing the interference signal as the error signal. The EOM that has high radio frequency to photonics conversion efficiency, is an excellent simulator of PAR. Four EOMs are independently driven by four clock-synchronized function generators with the same modulation frequency of 3 MHz, but the amplitude and phase are independently manipulated to generate four independent displacements on the squeezed phase quadrature [33]. The output of four homodyne detectors through the same four bandpass filters from 1.8 MHz to 4.5 MHz is demodulated with 3 MHz signals. The demodulation signals are clock synchronized with the corresponding return signals, but their phases can be independently manipulated to compensate for the delay difference of the processing circuit. The time-domain data from four mixers are collected and postprocessed to derive the estimated parameters under different weights, and demonstrate diverse quantum-enhanced sensing applications.
Figure 2.Experimental setup of quantum-enhanced PAR with a CV entangled network. OPA, optical parametric amplifier; HWP, half-wave plate; PBS, polarization beam splitter; BSN, beam splitter network; EOM, electro-optical modulator; HOM, homodyne detection. BPF, bandpass filter.
The noise variances of the measured PAR phase are limited by the initial squeezing factor and system loss. It is worth noting that the parametric amplification process (for the generation of a quadrature phase squeezed state) intensifies the noise coupling between the seed and pump field, which usually results in the degradation of the squeezing factor [41]. As the pump factor is increased, the noise coupling is enhanced. Thus, we need to make a trade-off between the pump factor and noise coupling, both of which are key factors affecting the squeezing factor. Here, the OPA operates at the pump factor of , with quantum noise reduction of generated, providing a superior quantum resource for the downstream experiment.
During the experiment, delicate mode-matching with interference efficiency of 99.8% is accomplished. The optimal weights are scheduled in terms of the different mission requirements, and the number of optical elements is as small as possible, reducing the additional noise coupling to the maximum extent and minimizing the estimation variance of diverse physical scenarios. On the other hand, the phase of the demodulation signal is carefully optimized to suppress the noise coupling of anti-squeezing quadrature into squeezing quadrature as much as possible.
Suppose that the PAR operates in single-transmitter and multi-receiver mode; we show the experimental results of quantum-enhanced PAR in two sensing tasks. PAR1 serves as a transmitter and receiver, and all other nodes act as receivers.
Under ideal conditions, three angles of arrival define a point in space, which is the principle of target positioning. More details can be found in Appendix A. However, the quantum noise of an optical field imposes a limitation on positioning precision, defining a space region instead of a point. Positioning precision can be improved by reducing the estimation variance of each angle of arrival. According to the above analysis, a quantum sensor network with an equally weighted distribution can offer a precision advantage in the angle of arrival estimation over the other weighted one. We estimate the average displacement for radio-frequency phase () by measuring the phase quadrature of each node and summing the photocurrents of four homodyne detectors. Figure 3(a) represents the estimation of average field amplitude and estimation variance for as a function of the applied PAR phases for all sensors synchronously. We repeatedly measured the dependence of estimated amplitude (average displacement) on applied PAR phase from zero to with phase interval of 20 deg. To reach optimum estimation precision, the applied PAR phase is manipulated to generate identical displacements on the phase quadrature in each sensor node simultaneously. By minimizing the estimation variance, a quantum noise reduction is achieved, confirming the entanglement properties of the sensor network with equally weighted distribution. At this point, the splitting ratios of three variable beam splitters (VBSs) are 25:75, 33:67, and 50:50, and the three angles of arrival are quantified. Figure 3(b) shows the measured noise powers for three phase differences for estimating the angle of arrival at different applied PAR phases. The measurement noise variances of three phase differences, represented by purple, green, and orange, are plotted for both quantum (square) and classical (circle) cases. The estimation variances for several cases are normalized to the shot noise limit. It is evident that the quantum-enhanced sensor network leads to a reduction in estimation variance, whereas such a noise reduction mechanism is absent without the quantum resource. The reduced noise powers of for , for , and for are obtained, which means the detection range is expanded by at least 12.8%. The noise variances are independent of the applied PAR phases, allowing the realization of 360-deg arbitrary scanning. The inset of Fig. 3(b) shows the principle of the improved positioning precision. It can be inferred that the uncertainty region of our protocol with multipartite entanglement is reduced to 0.4837 in three-dimensional space, in contrast with the classical protocol, corresponding to positioning precision improvement of 51.6%. Suppose we individually estimate each phase difference (, ) with the optimal weights 1/2:1/2:0:0 (1/2:0:1/2:0, 1/2:0:0:1/2); then we can obtain higher sensitivity improvement (6.8 dB) than that of simultaneous estimation using similar quantum states. However, the individual estimation will spend more time performing the measurement task (three times), which is inappropriate for the positioning protocol of a moving target.
Figure 3.Estimation of average field amplitude and normalized noise powers versus applied PAR phases. (a) Estimated average field amplitude. Data points: average displacement from homodyne detection; orange curves: sinusoidal fit; orange shaded area: estimation uncertainties for the entangled (dark color) and classical separable (light color) sensor networks. The blue curve shows the time-domain measurement result of standard quantum limit; the purple, light purple, green, and light green curves represent the measurement results with the entangled sensor network at several PAR phases:
3. QUANTUM RANGING PROTOCOL
In the quantum ranging case, except for target positioning, target ranging is another critical specification of a radar system, which can be expressed as (see Appendix B for more details)
Figure 4.Schematic illustration of quantum ranging based on CV entangled network.
To achieve optimum performance in quantum ranging, weighted distributions need to be configured on the grounds of the physical quantity measured. At the same time, the PAR network is arranged in a linear topology. The optimal weight :3::1/3 (2::4:) is arranged to implement the minimum estimation variance of the first (second) derivative of edge node PAR1; thereby, the splitting ratios of the VBSs correspond to 27.5:72.5, 62:38, and 82:18 (17:83, 50:50, and 80:20, respectively). The negative signs in the weights are introduced by adding a -phase delay in the corresponding optical path. Figure 5 shows the estimated average amplitude of the first and second derivatives of PAR1 as a function of applied PAR phase, with the shaded area representing the estimation uncertainty of the first (second) derivative of the node, dark orange (green) for the quantum sensing protocol, and light orange (green) for the classical protocol. To satisfy 360-deg fast scanning, a precision advantage of quantum ranging should be offered at an arbitrary azimuth angle, irrespective of the applied PAR phase. Compared with the classical protocol, the almost invariable noise reductions of for the first derivative and for the second derivative are implemented in the experiment, demonstrating the full scanning range with enhanced precision. The inset of Fig. 5 represents the topology structure of quantum ranging. In addition, for the noise suppression in three independent relative phase differences, the first and second derivatives of the edge node are still maintained when the applied PAR amplitude varies, which confirms that the PAR network can be compatible with the return signal with different intensities without sacrificing precision. It can be inferred from the results of Fig. 5 that the pitch angle with enhanced precision of 68.4% is achieved on all-around scanning. Furthermore, combined with noise suppression of 5.4 dB for , target ranging with enhanced precision of 69.3% can be realized. In other words, with the same signal to noise ratio (SNR) of the return signal, the detection range of radar is increased by 34.3% [42]. It is worth noting that quantum ranging can be realized only when both the first and second derivatives are measured simultaneously, meaning that at least three sensor nodes in line are required. In combination with quantum ranging, the space positioning of the target becomes more accurate.
Figure 5.Comparison of the estimated amplitude for the first and second derivatives of the PAR phase under two cases. Data points: means of the measured homodyne signals; curves: sinusoidal fit; shaded area: estimation uncertainties for entangled (dark color) and classical separable (light color) sensor networks. All signals are normalized using the same factor for standard quantum limit normalization.
4. CONCLUSION
In conclusion, we have implemented the demonstration of a quantum-enhanced PAR network exploiting a CV multipartite entanglement state, verifying the feasibility of quantum positioning and quantum ranging. By utilizing a sensor network with an average weighted distribution, quantum positioning offers a precision advantage of 51.6% in three-dimensional space over the classical scheme, increasing the detection range by 12.8%. Moreover, in virtue of diverse weighted distributions, quantum ranging with enhanced precision of 69.3% can be realized; thus, the detection range of radar is increased by 34.3% with the same SNR of the return signal. We expect to construct a distributed sensing network via a multipartite entanglement state, which can be applied to track moving targets in PAR. However, for realistic PAR, the frequency of a return signal from a target that is not constant depends on the speed of the moving target. Therefore, to satisfy diverse target tracking (different speeds of movement), we need the generation of a broad-bandwidth multipartite entanglement state and balanced homodyne detectors.
APPENDIX A: PRINCIPLE OF SPACE POSITIONING
Figure
Figure 6.Principle of quantum positioning based on quantum-enhanced sensor network.
For space positioning, the scanning surfaces from four sensor nodes interfere at the target position by manipulating the phase of the applied PAR field. Space positioning with angles of arrival has a significant advantage in less data and simple data processing since only the phase measurement and distance of each two PAR nodes are required.
APPENDIX B: THEORETICAL MODEL OF THE QUANTUM RANGING
Figure
Figure 7.Principle of quantum ranging based on quantum-enhanced sensor network.
From Eqs. (
The angle of arrival in Eq. (
Another key factor of quantum ranging is the derivative of the angle of arrival, expressed as
For given radar nodes, target ranging can be translated into the measurement of the angle of arrival and its first derivative of edge node PAR1. Taking Eqs. (
In summary, target positioning in virtue of the angle of arrival in three dimensions has relatively low precision. In combination with target ranging, the space positioning of the target becomes more accurate.
APPENDIX C: CALCULATION OF OPTIMUM WEIGHTS
In Appendix
References
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